\section{Contraction dynamics on flag manifolds}
\label{sec:contraction_appendix}

Let $V$ be a real vector space, and let $A_n$ be a sequence of
elements of $\PGL(V)$. It is sometimes possible to determine the
global dynamical behavior of $A_n$ on $\P(V)$ by considering the
action of $A_n$ on a small open subset of $\P(V)$: if there is an open
subset $U \subset \P(V)$ such that $A_n \cdot U$ converges to a point
in $\P(V)$, then in fact there is a dense open subset
$U_- \subset \P(V)$ (the complement of a hyperplane) on which $A_n$
converges to the same point, uniformly on compacts.

A similar statement holds for the action of $A_n$ on Grassmannians
$\Gr(k, V)$. These claims can be proved by considering the behavior
of the singular value gaps of $A_n$ as $n \to \infty$.

In this appendix we give a general result along these lines, where we
take sequences of group elements $g_n \in G$ for a semisimple Lie
group $G$ with no compact factor and trivial center, and consider the
limiting behavior of $g_n$ on open subsets of some flag manifold
$\pflags$, where $\pargroup$ is a parabolic subgroup.

\localContractingImpliesGlobalContracting*

We will prove \Cref{prop:local_contracting_implies_global_contracting}
by reducing it to the case where $G = \PGL(d,\R)$ and
$\pargroup = P_1$ is the stabilizer of
$[e_1] \in \RP^{d-1} \simeq G/P_1$. In this situation,
$\pargroup$-divergence can be understood in terms of the behavior of
the \emph{singular value gaps} of the sequence $g_n$:

\begin{prop}
  Suppose that $G = \PGL(d, \R)$, and let $\pargroup = P_1 \subset G$
  be the stabilizer of a line in $\R^d$. A sequence $g_n \in G$ is
  $P_1$-divergent if and only if
  \[
    \frac{\sigma_1(g_n)}{\sigma_2(g_n)} \to \infty,
  \]
  where $\sigma_i(g_n)$ is the $i$th-largest singular value of $g_n$.
\end{prop}

For convenience, we give a proof of
\Cref{prop:local_contracting_implies_global_contracting} in this
special case.
\begin{lem}
  \label{lem:local_contracting_implies_sv_gaps}
  Let $g_n$ be a sequence in $\PGL(d,\R)$, and suppose that for a
  nonempty open subset $U \subset \RP^{d-1}$, $g_nU$ converges to a
  point in $\RP^{d-1}$. Then, the singular value gap
  \[
    \frac{\sigma_1(g_n)}{\sigma_2(g_n)}
  \]
  tends to $\infty$ as $n \to \infty$.
\end{lem}
\begin{proof}
  It suffices to show that any subsequence of $g_n$ has a subsequence
  which satisfies the property. Using the Cartan decomposition of
  $\PGL(d,\R)$, we can write
  \[
    g_n = k_na_nk_n',
  \]
  for $k_n, k_n' \in K = \PO(d)$ and $a_n$ a diagonal matrix whose
  diagonal entries are $\sigma_1, \ldots, \sigma_d$. Up to subsequence
  $k_n$ and $k_n'$ converge respectively to $k, k' \in K$. For
  sufficiently large $n$, $k_n' U \cap k' U$ is nonempty, so by
  replacing $U$ with $k'U$ we can assume that $k_n' = \mr{id}$ for all
  $n$. Furthermore, if $k_na_n U$ converges to a point
  $z \in \RP^{d-1}$, then $a_nU$ converges to $k^{-1}z$.

  So, $a_nU$ converges to a point, and since $a_n$ is a diagonal
  matrix, the gap between the moduli of its largest and second-largest
  eigenvalues must be unbounded.
\end{proof}

To prove the general case of
\Cref{prop:local_contracting_implies_global_contracting}, we take an
irreducible representation $\zeta:G \to \PGL(V)$ coming from
\Cref{thm:flag_irrep_embedding}, so that $\pargroup$ maps to the
stabilizer of a line $\ell$ in $V$, $\oppgroup$ maps to the stabilizer
of a hyperplane $H$ in $V$, and $g\pargroup g^{-1}$,
$h\oppgroup h^{-1}$ are opposite if and only if $\zeta(g)\ell$,
$\zeta(h)H$ are transverse. As in \cref{sec:flag_convergence}, this
determines embeddings $\iota:\flags \to \P(V)$ and
$\iota^*:\oppflags \to \P(V^*)$ by
\[
  \iota(g\pargroup) = \zeta(g)\ell, \qquad \iota^*(g\oppgroup) =
  \zeta(g)H.
\]

The representation $\zeta$ additionally has the property that for any
sequence $g_n \in G$, the singular value gaps
\[
  \sigma_1(\zeta(g_n)) / \sigma_2(\zeta(g_n))
\]
are unbounded if and only if $g_n$ is $\pargroup$-divergent (see
\cite{ggkw2017anosov}, Lemma 3.7).

\begin{proof}[Proof of
  \Cref{prop:local_contracting_implies_global_contracting}]
  By \cite{zimmer2018proper}, Lemma 4.7, there exist flags
  $\xi_1, \ldots, \xi_D \in U$ so that lifts of $\iota(\xi_i)$ give a
  basis of $V$. Since $g_n \cdot U$ converges to a point in $\flags$,
  the set
  \[
    \{\zeta(g_n) \cdot \iota(\xi_i) : 1 \le i \le D\}
  \]
  converges to a single point in $\P(V)$.

  This means that we can fix lifts $\tilde{\iota(\xi_i)} \in V$ so
  that, up to a subsequence, $\zeta(g_n)$ takes the projective
  $(D-1)$-simplex
  \[
    \left[\sum_{i=1}^D \lambda_i \tilde{\iota(\xi_i)} : \lambda_i >
      0\right] \subset \P(V)
  \]
  to a point. This simplex is an open subset of $\P(V)$. Now we can
  apply \Cref{lem:local_contracting_implies_sv_gaps} to see that the
  sequence $g_n$ is $\pargroup$-divergent.

  We now just need to check that $\xi$ is the unique $P^+$-limit point
  of $g_n$. Choose any subsequence of $g_n$. Then any
  $P^+$-contracting subsequence $g_m$ of this subsequence satisfies
  \[
    g_m|_{\Opp(\xi_-)} \to \xi'
  \]
  uniformly on compacts for some $\xi_- \in \oppflags$ and
  $\xi' \in \pflags$. But since $\Opp(\xi_-)$ is open and dense, it
  intersects $U$ nontrivially and thus $\xi' = \xi$.
\end{proof}


\subsection{Compatible systems of open sets}

A relative quasigeodesic automaton always exists for any relatively
hyperbolic group (although we will not prove this fact in full
generality). We will give a way to construct a relative quasigeodesic
automaton using the \emph{convergence group dynamics} of a group
acting on its Bowditch boundary.

\begin{definition}
  Suppose that $\Gamma$ acts on a metrizable space $M$ by
  homeomorphisms, and let $\mc{G}$ be a $\Gamma$-graph. A
  \emph{$\mc{G}$-compatible system of open sets} for the action of
  $\Gamma$ on $M$ is an assignment of an open subset $U_v \subset M$
  to each vertex $v$ of $\mc{G}$ such that for each edge $e = (v,w)$
  in $\mc{G}$, for some $\eps > 0$, we have
  \begin{equation}
    \label{eq:compatibility}
    \alpha \cdot N_M(U_w, \eps)  \subset U_v
  \end{equation}
  for all $\alpha \in T_v$.
\end{definition}

\begin{remark}
  If $\mc{G}$ has no parabolic vertices (so each set $T_v$ contains a
  single group element $\alpha_v \in \Gamma$), then
  (\ref{eq:compatibility}) is equivalent to requiring
  $\alpha_v \cdot \overline{U_w} \subset U_v$ for every edge $(v, w)$
  in $\mc{G}$. When $\mc{G}$ has parabolic vertices (so $T_v$ may be
  infinite), (\ref{eq:compatibility}) is in general a stronger
  condition.
\end{remark}

\begin{prop}
  \label{prop:bowditch_graphs_geodesic}
  Let $\mc{G}$ be a $\Gamma$-graph, and let
  $\{U_v : v \textrm{ vertex of \mc{G}}\}$ be a $\mc{G}$-compatible
  system of subsets of $\bgamh$ for the action of
  $\Gamma$ on $\bgamh$.

  There is a constant $D > 0$ satisfying the following: let
  $\{\alpha_n\}$ be an infinite $\mc{G}$-path, corresponding to a
  vertex path $\{v_n\}$, and suppose the sequence
  $\{\gamma_n = \alpha_1 \cdots \alpha_n\}$ is divergent in
  $\Gamma$. Then for any point $z$ in the intersection
  \[
    U_\infty = \bigcap_{n=1}^\infty \alpha_1 \cdots \alpha_n U_{v_{n+1}},
  \]
  the sequence $\gamma_n$ lies within Hausdorff distance $D$ of a
  geodesic ray in $\conedgraph$ tending towards $z$.
\end{prop}
\begin{proof}
  Fix a point $z \in U_\infty$, and write $z = z_+$ and
  $U_n = U_{v_n}$. We first claim that there is a uniform $\eps > 0$
  and a point $z_- \in \bgamh$ such that
  \begin{equation}
    \label{eq:endpoints_not_close}
    d(\gamma_n^{-1} z_+, \gamma_n^{-1} z_-) > \eps
  \end{equation}
  for all $n \ge 0$.

  To prove the claim, choose a uniform $\eps > 0$ so that for every
  vertex $v$ in $\mc{G}$, we have
  $N(U_v, \eps) \ne \bgamh$, and for every edge $(v, w)$
  in $\mc{G}$ and every $\alpha \in T_v$, we have
  $\alpha \cdot N(U_w, \eps) \subset U_v$. Then we choose some
  $z_- \in \bgamh - \overline{N}(U_1, \eps)$.

  By the $\mc{G}$-compatibility condition, we know that for any $n$,
  $\gamma_nU_{n+1} \subset \ldots \subset \gamma_1U_2 \subset U_1$, so
  we know that $d(z_+, z_-) > \eps$.
  
  Then, for any $n \ge 1$, we have
  \[
    \gamma_n^{-1} z_+ \in U_{n+1}.
  \]
  Moreover since $\gamma_n N(U_{n+1}, \eps) \subset U_1$, we also have
  \[
    \gamma_n^{-1} z_- \in \bgamh - N(U_{n+1}, \eps).
  \]
  So for all $n$ we have
  $d(\gamma_n^{-1}z_+, \gamma_n^{-1}z_-) > \eps$, which establishes
  that \eqref{eq:endpoints_not_close} holds for all $n$.

  Now, consider a bi-infinite geodesic $c$ in a Gromov model $Y$ for
  $\Gamma$ joining $z_+$ and $z_-$. The sequence of geodesics
  $\gamma_n^{-1} \cdot c$ has endpoints in
  $\dee Y = \bgamh$ lying distance at least $\eps$
  apart, so each geodesic in the sequence passes within a uniformly
  bounded neighborhood of a fixed basepoint $y_0 \in Y$. Therefore
  $\gamma_n \cdot y_0$ lies in a uniformly bounded neighbood of the
  geodesic $c$.

  Since $\gamma_n$ is divergent, $\gamma_n y_0$ can only accumulate at
  either $z_+$ or $z_-$. But in fact $\gamma_n y_0$ can only
  accumulate at $z_+$---for in the construction of $c$ above, we could
  have chosen any $z_-$ in the nonempty open set
  $\bgamh - \overline{N}(U_1, \eps)$, and since
  $\bgamh$ is perfect there is at least one such
  $z_-' \ne z_-$.

  This implies that $\gamma_n$ is a conical limit sequence in
  $\Gamma$, limiting to $z_+$. Since the distance between $\gamma_n$
  and $\gamma_{n+1}$ is bounded in $\conedgraph$, the desired
  conclusion follows.
\end{proof}

\begin{definition}
  Let $\mc{G}$ be a $\Gamma$-graph. An infinite $\mc{G}$-path
  $\{\alpha_n\}$ is \emph{divergent} if the sequence
  $\{\gamma_n = \alpha_1 \cdots \alpha_n\}$ leaves every bounded
  subset of $\Gamma$.

  We say that a $\Gamma$-graph $\mc{G}$ is divergent if \emph{every}
  infinite $\mc{G}$-path is divergent.
\end{definition}





Whenever $\{U_v\}$ is a $\mc{G}$-compatible system of open sets for a
$\Gamma$-graph $\mc{G}$, one can think of a $\mc{G}$-path
$\{\alpha_n\}$ as giving a \emph{symbolic coding} of a point in the
intersection
\[
  \alpha_1 \cdots \alpha_n U_{n+1}.
\]
The following proposition gives a way to \emph{construct} such a
coding for a given point $z \in \bgamh$, given an
appropriate pair of open coverings of the Bowditch boundary
$\bgamh$ compatible with a $\Gamma$-graph $\mc{G}$.

\begin{prop}
  \label{prop:relative_geodesic_automaton_from_bb}
  Let $\mc{G}$ be a \emph{divergent} $\Gamma$-graph. Suppose that for
  each vertex $a \in \mc{G}$, there exist open subsets $V_a, W_a$ of
  $\bgamh$ such that the following conditions hold:
  \begin{enumerate}
  \item The sets $\{W_a\}$ give a $\mc{G}$-compatible system of sets
    for the action of $\Gamma$ on $\bgamh$.
  \item For all vertices $a$, we have $V_a \subset W_a$ and
    $\overline{W_a} \ne \bgamh$.
  \item The sets $V_a$ give an open covering of $\bgamh$.
  \item \label{item:conical_edge_exists} For every $z \in \bgamh$
    and every non-parabolic vertex $a$ such that $z \in V_a$, there is
    an edge $(a, b)$ in $\mc{G}$ such that
    $\alpha_a^{-1} \cdot z \in V_b$ for $\{\alpha_a\} = T_a$.
  \item \label{item:parabolic_edge_exists} For every
    $z \in \bgamh$ and every \emph{parabolic} vertex $a$ such
    that $z \in V_a - \{p_a\}$, there is an edge $(a, b)$ in $\mc{G}$
    and $\alpha \in T_a$ such that $\alpha^{-1} \cdot z \in V_b$.
  \end{enumerate}
  Then $\mc{G}$ is a relative quasigeodesic automaton for $\Gamma$.
\end{prop}

\begin{figure}[h]
  \centering
 
  \def14cm{8cm}
  \import{svg-inkscape/}{automaton_contract_svg-tex.pdf_tex}
  \caption{Illustration for the proof of
    \Cref{prop:relative_geodesic_automaton_from_bb}. The group element
    $\alpha_n$ nests an $\eps$-neighborhood of $W_{a_{n+1}}$ inside of
    $W_{a_n}$ whenever $\alpha_n \cdot V_{a_{n+1}}$ intersects
    $V_{a_n}$.}
  \label{fig:g_compatible_inclusion}
\end{figure}
\begin{figure}[h]
  \centering
 
  \def14cm{.8\textwidth}
  \import{svg-inkscape/}{nesting_maps_3_svg-tex.pdf_tex}
  \caption{By iterating the nesting procedure backwards, we produce an
    infinite $\mc{G}$-path and a sequence of subsets intersecting in
    the initial point $z = z_0$.}
  \label{fig:nested_subsets}
\end{figure}
\begin{proof}
  \Cref{prop:bowditch_graphs_geodesic} implies that any infinite
  $\mc{G}$-path lies finite Hausdorff distance from a geodesic ray in
  $\conedgraph$. So, we just need to show that every
  $z \in \bgamh$ is the limit of a $\mc{G}$-path.

  The idea behind the proof is to use the fact that the sets $V_a$
  cover $\bgamh$ to show that we can keep ``expanding''
  a neighborhood of $z$ in $\bgamh$ to construct a path
  in $\mc{G}$ limiting to $z$. The $\{V_a\}$ covering tells us how to
  find the next edge in the path, and the $\{W_a\}$ cover gives us the
  $\mc{G}$-compatible system we need to show that the path is a
  geodesic. 

  We let $A$ denote the vertex set of $\mc{G}$. When $a \in A$ is not
  a parabolic vertex, we write $T_a = \{\gamma_a\}$.

  \begin{aside}{Case 1: $z$ is a conical limit point}
    Fix $a \in A$ so that $z \in V_a$. We take $z_0 = z$, $a_0 = a$,
    and define sequences
    $\{z_n\}_{n=0}^\infty \subset \conbdry \Gamma$,
    $\{a_n\}_{n=0}^\infty \subset A$, and
    $\{\alpha_n\}_{n=1}^\infty \subset \Gamma$ as follows:
    \begin{itemize}
    \item If $a_n$ is not a parabolic vertex, then we choose
      $\alpha_{n+1} = \gamma_{a_n}$. Let
      $z_{n+1} = \alpha_{n+1}^{-1} \cdot z_n$. Since conical limit
      points are invariant under the action of $\Gamma$, $z_{n+1}$ is
      a conical limit point. By condition
      \ref{item:conical_edge_exists}, there is a vertex $a_{n+1}$
      satisfying $z_{n+1} \in V_{a_{n+1}}$ with $(a_n, a_{n+1})$ an
      edge in $\mc{G}$.
    \item If $a_n$ is a parabolic vertex, then since $z_n$ is a
      conical limit point, $z_n \ne p$ for $p = p_{a_n}$. Then
      condition \ref{item:parabolic_edge_exists} implies that there
      exists some $\alpha_{n+1} \in T_{a_n}$ so that
      $\alpha_{n+1}^{-1} \cdot z_n \in V_{a_{n+1}}$ for an edge
      $(a_n, a_{n+1})$ in $\mc{G}$. Again,
      $z_{n+1} = \alpha_{n+1}^{-1} \cdot z_n$ must be a conical limit
      point since $\conbdry\Gamma$ is $\Gamma$-invariant.
    \end{itemize}

    The sequence $\{\alpha_n\}$ necessarily gives a $\mc{G}$-path. By
    assumption the sequence
    \[
      \gamma_n = \alpha_1 \cdots \alpha_n
    \]
    is divergent. And by construction $z = \gamma_nz_n$ lies in
    $\gamma_n W_{a_n}$ for all $n$. So,
    \Cref{prop:bowditch_graphs_geodesic} implies that $\gamma_{n}$ is
    a conical limit sequence, limiting to $z$. See
    \Cref{fig:nested_subsets}.
  \end{aside}
  \begin{aside}{Case 2: $z$ is a parabolic point}
    As before fix $a \in A$ so that $z \in V_a$, and take $z_0 = z$,
    $a_0 = a$. We inductively define sequences $z_n$, $a_n$,
    $\alpha_n$ as before, but we claim that for some finite $N$, $a_N$
    is a parabolic vertex with $z_N = p_{a_N}$. For if not, we can
    build an infinite $\mc{G}$-path (as in the previous case) limiting
    to $z$. But then, \Cref{prop:bowditch_graphs_geodesic} would imply
    that $z$ is actually a conical limit point. So, we must have
    \[
      z = \gamma_N a_N = \alpha_1 \cdots \alpha_N a_N
    \]
    as required.
  \end{aside}
\end{proof}

\begin{remark}
  In a typical application of
  \Cref{prop:relative_geodesic_automaton_from_bb}, it will not be
  possible to construct the open coverings $\{V_a\}$ and $\{W_a\}$ so
  that $V_a = W_a$ for all vertices $a$. In particular we expect this
  to be impossible whenever $\bgamh$ is connected.
\end{remark}


To conclude this section, we make one more observation about systems
of $\mc{G}$-compatible sets as in
\Cref{prop:relative_geodesic_automaton_from_bb}.

\begin{lem}
  \label{lem:close_points_share_prefixes}
  Let $\Gamma$ be a relatively hyperbolic group, let $\mc{G}$ be a
  $\Gamma$-graph, and let $\{V_a\}$, $\{W_a\}$ be an assignment of
  open subsets of $\bgamh$ to vertices of $\mc{G}$ satisfying the
  hypotheses of \Cref{prop:relative_geodesic_automaton_from_bb}.

  Fix $z \in \conbdry \Gamma$ and $N \in \N$. There exists
  $\delta > 0$ so that if $d(z, z') < \delta$, then there are
  $\mc{G}$-paths $\{\alpha_n\}, \{\beta_n\}$ limiting to $z, z'$
  respectively, with $\alpha_i = \beta_i$ for all $i < N$.
\end{lem}
\begin{proof}
  Let $\{\alpha_n\}$ be a $\mc{G}$-path limiting to $z$ coming from
  the construction in \Cref{prop:relative_geodesic_automaton_from_bb},
  passing through vertices $v_n$. We choose $\delta > 0$ small enough
  so that if $d(z, z') < \delta$, then $z'$ lies in the set
  \[
    \alpha_1 \cdots \alpha_N V_{v_{n+1}}.
  \]
  Then for every $i < N$, we have
  \[
    \alpha_i^{-1} \alpha_{i-1}^{-1} \cdots \alpha_1^{-1} z' \in
    V_{v_{i+1}}.
  \]
  As in \Cref{prop:relative_geodesic_automaton_from_bb}, we can then
  extend $\{\alpha_n\}_{n=1}^{N-1}$ to a $\mc{G}$-path limiting to
  $z'$.
\end{proof}


\subsection{Convex projective structures}

Let $V$ be a real $(d+1)$-dimensional vector space. Recall that a
subset $\Omega \subset \P(V)$ is \emph{properly convex} if
$\overline{\Omega}$ is a convex subset of an affine chart in
$\P(V)$. A manifold $M$ (possibly with boundary) has a \emph{convex
  projective structure} if $M$ can be realized as a quotient
$\Omega / \Gamma$ for a discrete subgroup
$\Gamma \subseteq \Aut(\Omega)$, where
\[
  \Aut(\Omega) = \{\gamma \in \PGL(V) : \gamma \cdot \Omega =
  \Omega\}.
\]

When $\Omega$ is open, we say it is a \emph{properly convex
  domain}. Then any manifold $M = \Omega / \Gamma$ has empty
boundary. We only consider group actions on properly convex domains in
this section, but in \cref{sec:generalized_cusps}, we will only ask
for $\Omega$ to have nonempty interior.

\subsubsection{Faces in convex domains}

If $\Omega \subset \P(V)$ is a properly convex domain, a \emph{face}
of $\Omega$ is an equivalence class in $\dee \Omega$ under the
relation which identifies distinct points $x,y \in \dee \Omega$ if
there is an open projective line segment in $\dee \Omega$ containing
both $x$ and $y$.

If $x \in \dee \Omega$, we let $F_\Omega(x)$ denote the unique face of
$\Omega$ containing $x$. The \emph{support} of a face $F$, denoted
$\mathrm{supp}(F)$, is the projective span $[\spn(F)] \subset
\P(V)$. The \emph{boundary} of a face $\dee F$ is its boundary viewed
as a convex open subset of $\mathrm{supp}(F)$.

\subsubsection{The Hilbert metric}

Whenever $\Omega$ is a properly convex domain, one can define the
\emph{Hilbert metric} on $\Omega$ as follows:
\begin{definition}
  \label{defn:hilbert_metric}
  For a pair of distinct points $x, y \in \Omega$, we let
  \[
    d_\Omega(x, y) = \frac{1}{2}\log[u, v; x, y],
  \]
  where $u, v$ are the two points in $\dee \Omega$ such that
  $u, x, y, v$ lie on a projective line in that order, and
  $[a, b; c, d]$ is the \emph{cross-ratio}
  \[
    [a, b; c, d] = \frac{(d - a) (c - b)}{(c - a) (d - b)}.
  \]
  Here the differences can be measured under any affine identification
  of the projective line spanned by $x, y$ with $\R \cup \{\infty\}$.

  For $u \in \Omega$ and $r > 0$, we let $B_\Omega(u, r)$ denote the
  open ball in $(\Omega, d_\Omega)$ of radius $r$ about $u$.
\end{definition}
$(\Omega, d_\Omega)$ is a proper geodesic metric space, on which
$\Aut(\Omega)$ acts by isometries. This means that $\Aut(\Omega)$ acts
properly on $\Omega$---in particular, discrete subgroups of
$\Aut(\Omega)$ act properly discontinuously.

The Hilbert metric gives us another perspective on the faces of
$\Omega$. Each face $F$ of $\Omega$ is a properly convex subset of
$\P(V)$, open in its own projective span. This allows us to define a
restricted Hilbert metric $d_F$ on $F$.

The proposition below is a standard result in the theory of convex
projective structures, and is a direct consequence of the definition
of the Hilbert metric.
\begin{prop}
  \label{prop:hilbert_metric_faces}
  Let $\Omega$ be a properly convex domain, let $F$ be a face of
  $\Omega$, and let $x_n$ be a sequence in $\Omega$ converging to some
  $x_\infty \in F$.

  For any $D > 0$, if $y_n \in \Omega$ is a sequence satisfying
  \[
    d_\Omega(x_n, y_n) \le D,
  \]
  then any accumulation point $y_\infty$ of $y_n$ lies in $F$, and
  \[
    d_F(x_\infty, y_\infty) \le D.
  \]
\end{prop}

\subsubsection{Dynamics of $\Aut(\Omega)$}

When $G$ is a semisimple Lie group, we say that a sequence $g_n \in G$
is \emph{divergent} if it leaves every compact subset of $G$. A
divergent sequence always has a $P^+$-divergent subsequence for some
parabolic $P^+ \subset G$. By taking a further subsequence, we can
find \emph{$P^\pm$-limit points} (see \Cref{defn:limit_set}) for $g_n$
and $g_n^{-1}$ in the flag manifolds $\pflags$ and $\oppflags$, for
$P^-$ opposite to $P^+$.

When $G = \PGL(V)$, we can interpret this in terms of \emph{attracting
  and repelling subspaces}.
\begin{prop}
  \label{prop:attracting_repelling_subspaces}
  Let $g_n \in \PGL(V)$ be a divergent sequence. Up to subsequence,
  there exist subspaces $E_+$, $E_-$ in $V$ such that
  \[
    \dim(E_+) + \dim(E_-) = \dim(V),
  \]
  and for any open set $U \subset \P(V)$ containing $[E_+]$ and any
  compact set $K \subset \P(V) - [E_-]$, for all sufficiently large
  $n$ we have
  \[
    g_n \cdot K \subset U.
  \]
\end{prop}
\begin{proof}
  For each $k$ with $1 \le k < d$, we let $P_k \subset \PGL(V)$ be the
  subgroup preserving a $k$-dimensional subspace of $V$. The maximal
  parabolic subgroups of $\PGL(V)$ are exactly the subgroups conjugate
  to some $P_k$. So any divergent sequence in $\PGL(V)$ is
  $P_k$-divergent for some $k$, and the result follows.
\end{proof}
The subspaces $E_+$, $E_-$ are respectively referred to as
\emph{attracting} and \emph{repelling} subspaces. They are uniquely
determined once a subsequence of $g_n$ is chosen and their dimensions
are specified. They are \emph{not} necessarily transverse.

The faces of $\Omega$ inform the limiting dynamical behavior of
divergent sequences in $\Aut(\Omega)$.

\begin{prop}
  \label{prop:attracting_spaces_in_faces}
  Let $\gamma_n$ be a divergent sequence in $\Aut(\Omega)$ for a
  properly convex domain $\Omega$, and suppose that for some
  $x \in \Omega$, the sequence $\gamma_n x$ accumulates on a face
  $F_+$ of $\Omega$. Then, after extracting a subsequence, there is an
  attracting subspace $E_+$ of $\gamma_n$ such that
  $[E_+] \subseteq \supp(F_+)$.
\end{prop}
\begin{proof}
  Let $x_\infty \in F_+$ be an accumulation point of the point of the
  sequence $\gamma_n x$. Using a diagonal argument, we can replace
  $\gamma_n$ with a subsequence so that for every $y$ in a countable
  dense subset of $B_\Omega(x,1)$, the sequence $\gamma_n \cdot y$ has
  a well-defined limit in the compact space
  $\P(V)$. \Cref{prop:hilbert_metric_faces} then implies that for
  every point $y \in B_\Omega(x,1)$, the sequence $\gamma_n y$
  converges to a unique point in $F_+$.

  Let $B_\infty$ be the set of accumulation points of
  $\gamma_n \cdot B_\Omega(x, 1)$, and let $W_\infty$ be the subspace
  $\spn(B_\infty) \subset V$.

  \Cref{prop:hilbert_metric_faces} implies that $B_\infty$ is a subset
  of the face $F = F_\Omega(x_\infty)$, so $[W_\infty]$ is a
  projective subspace of $\supp(F)$. Let $k = \dim(W_\infty)$. We
  claim that there is an open subset $U$ of the Grassmannian
  $\Gr(k, V)$ so that
  \[
    \gamma_n \cdot U \to \{W_\infty\}.
  \]
  This implies the desired result by
  \Cref{prop:local_contracting_implies_global_contracting}.

  To see the claim, fix $k$ points
  $z_1, \ldots, z_k \in B_\Omega(x, 1)$ so that the limits of the
  sequences $\gamma_n z_1, \ldots \gamma_n z_k$ span the projective
  subspace $[W_\infty]$. \Cref{prop:hilbert_metric_faces} implies that
  for some fixed $\eps > 0$, if $z_i'$ lies in $B_\Omega(z_i, \eps)$,
  then the limits of the sequences
  \[
    \gamma_n z_1', \ldots, \gamma_nz_k'
  \]
  are in general position, and therefore also span $[W_\infty]$.

  Then, if $U$ is the open set
  \[
    \{W \in \Gr(k, V) : W = u_1 \oplus \ldots \oplus u_k, [u_i] \in
    B_\Omega(z_i, \eps)\},
  \]
  we have that $\gamma_n U \to \{W_\infty\}$, as required.
\end{proof}

\subsection{Relatively hyperbolic convex cocompact groups}

Inspired by the theory of convex cocompact groups in rank-one, in
\cite{dgk2017convex}, Danciger-Gu\'eritaud-Kassel introduce a notion
of \emph{convex cocompact} projective structure:
\begin{definition}[\cite{dgk2017convex}, Definitions 1.10 and 1.11]
  Let $\Omega$ be a properly convex domain in $\P(V)$, and let
  $\Gamma \subseteq \Aut(\Omega)$ be discrete.

  \begin{enumerate}
  \item The \emph{full orbital limit set} $\Lambda_\Omega(\Gamma)$ of
    $\Gamma$ is the union (over all $x \in \Omega$) of the set of
    accumulation points in $\dee \Omega$ of $\Gamma \cdot x$.
  \item The group $\Gamma$ \emph{acts convex cocompactly on $\Omega$}
    if the convex hull of $\Lambda_\Omega(\Gamma)$ is nonempty, and
    $\Gamma$ acts cocompactly on the convex hull of
    $\Lambda_\Omega(\Gamma)$ in $\Omega$.
  \item $\Gamma$ is \emph{convex cocompact in $\P(V)$} if it acts
    convex cocompactly on some properly convex domain
    $\Omega \subset \P(V)$.
  \end{enumerate}
\end{definition}

Danciger-Gu\'eritaud-Kassel show that if $\Gamma$ is a hyperbolic
group acting convex cocompactly in $\P(V)$, then the inclusion
$\Gamma \hookrightarrow \PGL(V)$ is a $P_1$-Anosov representation,
where $P_1$ is the stabilizer of a point in $\P(V)$.

In \cite{weisman2020dynamical}, we showed that any group acting convex
cocompactly on some domain $\Omega \subset \P(V)$ acts with
``Anosov-like'' expansion dynamics on the faces of $\dee \Omega$. We
further showed that, when $(\Gamma, \mc{H})$ is a relatively
hyperbolic pair, $\Gamma$ acts convex cocompactly on $\Omega$, and
each $H \in \mc{H}$ also acts convex cocompactly on $\Omega$, then
there is an equivariant embedding of the Bowditch boundary $\bgamh$
into a \emph{quotient} of $\dee \Omega$.

In subsequent work \cite{iz2022structure}, Islam-Zimmer showed that
one does not need to assume that each $H \in \mc{H}$ acts convex
cocompactly on $\Omega$: it turns out that this follows automatically
from the fact that $\Gamma$ acts convex cocompactly. Together with our
earlier work, this gives the following:
\begin{theorem}[See \cite{weisman2020dynamical}, Theorem 1.16 and
  \cite{iz2022structure}, Theorem 1.6]
  \label{thm:convex_cocompact_boundary_embedding}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and suppose
  $\rho(\Gamma)$ acts convex cocompactly on a properly convex domain
  $\Omega$. Then there is an equivariant homeomorphism
  \[
    \psi:\bgamh \to \Lambda_\Omega(\Gamma) / \sim,
  \]
  where $x \sim y$ if $x,y \in \Lambda_\Omega(H)$ for some
  $H \in \mc{H}$.
\end{theorem}

\begin{remark}
  Islam-Zimmer also proved a version of
  \Cref{thm:convex_cocompact_boundary_embedding} in the more general
  context of \emph{naive} convex cocompact group actions, which we do
  not discuss here.
\end{remark}

Our goal now is to prove the following theorem, which shows that these
relatively hyperbolic convex cocompact groups fit into the framework
of EGF representations.

\convexCocompactBoundaryDynamics*

In order to prove \Cref{thm:convex_cocompact_boundary_dynamics}, we
will need a slightly different perspective on the full orbital limit
set of a discrete group $\Gamma \subseteq \Aut(\Omega)$.

\begin{definition}
  Let $\Omega$ be a convex projective domain. The \emph{dual domain}
  $\Omega^* \subset \P(V^*)$ is the set
  \[
    \Omega^* = \{w \in \P(V^*) : \hyp{w} \cap \overline{\Omega} =
    \emptyset\}.
  \]
\end{definition}

If $\Gamma \subset \PGL(V)$ is a subgroup of $\Aut(\Omega)$, then its
dual $\Gamma^* \subset \PGL(V^*)$ is a subgroup of
$\Aut(\Omega^*)$. If $\Gamma$ is convex cocompact in $\P(V)$, then
\cite{dgk2017convex}, Proposition 5.6 implies that there is some
$\Gamma$-invariant domain $\Omega$ so that $\Gamma$ acts convex
cocompactly on $\Omega$, and $\Gamma^*$ acts convex cocompactly on
$\Omega^*$.

\begin{definition}
  Let $\Gamma \subseteq \Aut(\Omega)$.

  \begin{enumerate}
  \item The \emph{dual full orbital limit set}
    $\Lambda_\Omega^*(\Gamma)$ is the full orbital limit set of
    $\Gamma^*$ in $\dee \Omega^*$.
  \item
    The \emph{flag-valued full orbital limit set}
    $\hat{\Lambda}_\Omega(\Gamma)$ is the set
    \[
      \hat{\Lambda}_\Omega(\Gamma) := \{(x, w) \in \projflags(V) : x
      \in \Lambda_\Omega(\Gamma), w \in \Lambda^*_\Omega(\Gamma)\}.
    \]
  \item The \emph{maximal domain} $\maxdomain(\Gamma)$ is the unique
    connected component of
    \[
      \P(V) - \bigcup_{w \in \Lambda_\Omega^*(\Gamma)} \hyp{w}
    \]
    containing $\Omega$. Equivalently, $\maxdomain(\Gamma)$ is the
    dual of the convex hull of $\Lambda_\Omega^*(\Gamma)$ in
    $\Omega^*$.
  \end{enumerate}
\end{definition}

We emphasize that $\maxdomain(\Gamma)$ is \emph{not} necessarily a
properly convex set, which means that we cannot always define a
Hilbert metric on it (so we do not have a guarantee that $\Gamma$ acts
properly discontinuously in general). However, when $\Gamma$ acts
convex cocompactly on $\Omega$, we do get a properly discontinuous
action, thanks to the following argument suggested by Jeff Danciger
and Fanny Kassel:

\begin{prop}
  \label{prop:maxdomain_accumulation}
  Let $\Gamma$ act convex cocompactly on a properly convex comain
  $\Omega$. For any $x \in \maxdomain(\Gamma)$, every accumulation
  point of $\Gamma \cdot x$ lies in $\Lambda_\Omega(\Gamma)$. In
  particular, $\Gamma$ acts properly discontinuously on
  $\maxdomain(\Gamma)$.
\end{prop}
\begin{proof}
  When $\maxdomain(\Gamma)$ is a \emph{properly} convex domain, this
  follows immediately from Proposition 4.18 in \cite{dgk2017convex},
  which says that whenever $\Gamma$ acts convex cocompactly on some
  domain $\Omega$, and $\Omega'$ is any $\Gamma$-invariant properly
  convex domain containing $\Omega$, then $\Gamma$ acts convex
  cocompactly on $\Omega'$ and
  $\Lambda_\Omega(\Gamma) = \Lambda_{\Omega'}(\Gamma)$.

  So, we consider the case where $\maxdomain(\Gamma)$ is \emph{not}
  properly convex. We may assume our domain $\Omega$ is chosen so that
  $\Gamma$ acts convex cocompactly on both $\Omega$ and
  $\Omega^* \subset \P(V^*)$. Since $\maxdomain(\Gamma)$ is not
  properly convex, its dual $\maxdomain(\Gamma)^*$ (given by the
  convex hull of $\Lambda^*_\Omega(\Gamma)$ in $\Omega^*$) has empty
  interior (i.e. it spans a proper projective subspace of $\P(V^*)$).

  Given any $\eps > 0$, we let $\Omega_\eps^*$ be the uniform
  $\eps$-neighborhood of $\maxdomain(\Gamma)^*$, with respect to the
  Hilbert metric on $\Omega^*$. We let $\Omega_\eps \subset \P(V)$
  denote the dual of $\Omega_\eps^*$. Note that $\Omega_\eps$ is a
  $\Gamma$-invariant properly convex subset of $\maxdomain(\Gamma)$,
  containing $\Omega$.

  Since duality reverses inclusions, and the intersection
  \[
    \bigcap_{\eps \to 0} \Omega_\eps^*
  \]
  is exactly the set $\maxdomain(\Gamma)^*$, the union
  \[
    \bigcup_{\eps \to 0} \Omega_\eps
  \]
  is the set $\maxdomain(\Gamma)$. So, if we fix
  $x \in \maxdomain(\Gamma)$, for some $\eps > 0$ we have
  $x \in \Omega_\eps$. Then we apply Proposition 4.18 in
  \cite{dgk2017convex} to the \emph{properly} convex domain
  $\Omega_\eps$ to see that $\Gamma \cdot x$ must accumulate in the
  set $\Lambda_\Omega(\Gamma)$.
\end{proof}

We are almost ready to prove
\Cref{thm:convex_cocompact_boundary_dynamics}. We quote the following
observation from \cite{weisman2020dynamical}:
\begin{prop}[\cite{weisman2020dynamical}, Proposition 8.1]
  \label{prop:segments_in_peripherals}
  Let $\Gamma \subset \PGL(V)$ act convex cocompactly on a properly
  convex domain $\Omega$. If $\Gamma$ is hyperbolic relative to a
  collection of subgroups $\mc{H}$ also acting convex cocompactly on
  $\Omega$, then every nontrivial projective segment in
  $\Lambda_\Omega(\Gamma)$ is contained in $\Lambda_\Omega(H)$ for
  some $H \in \mc{H}$.
\end{prop}

Using a theorem of Danciger-Gu\'eritaud-Kassel (\cite{dgk2017convex},
Theorem 1.18), one can strengthen this result:
\begin{cor}
  \label{cor:segments_in_peripherals}
  In the context of \Cref{prop:segments_in_peripherals}, it is
  possible to choose the convex domain $\Omega$ so that every
  nontrivial projective segment in $\dee \Omega$ is contained in
  $\Lambda_\Omega(H)$ for some $H \in \mc{H}$.
\end{cor}

We also observe:
\begin{prop}
  \label{prop:boundary_embedding_conical_pts}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, let
  $\rho:\Gamma \to \PGL(V)$ be representation such that $\rho(\Gamma)$
  acts convex cocompactly on a domain $\Omega$, and let
  $\psi:\bgamh \to \Lambda_\Omega(\Gamma) / \sim$ be the map
  coming from \Cref{thm:convex_cocompact_boundary_embedding}.

  If $z \in \bgamh$ is a conical limit point, and $\gamma_n$ is a
  sequence limiting to $z$ in
  $\overline{\Gamma} = \Gamma \sqcup \bgamh$, then $\psi(z) \in \P(V)$
  is the unique one-dimensional attracting subspace for
  $\rho(\gamma_n)$ in $\P(V)$.
\end{prop}
\begin{proof}
  It suffices to show that any subsequence of $\gamma_n$ has a further
  subsequence which has $\psi(z)$ as a one-dimensional attracting
  subspace. So, using the convergence group property, we can take a
  subsequence and assume that there is some point $y \in \bgamh$ so
  that $\gamma_n$ converges to $z$ on every point in the set
  $\bgamh - \{y\}$. We can further assume that the pair
  $(\Gamma, \mc{H})$ is not elementary, so $\bgamh$ contains
  infinitely many points.

  So, by \Cref{cor:segments_in_peripherals}, we can find distinct
  points $u, v \in \Lambda_\Omega(\Gamma)$ so that the projective line
  segment $(u,v)$ lies in $\Omega$, and $\rho(\gamma_n)u$,
  $\rho(\gamma_n)v$ both converge to $\psi(z)$. Corollary 4.10 in
  \cite{dgk2017convex} implies that
  $F_\Omega(\psi(z)) \subset \Lambda_\Omega(\Gamma)$, and then
  \Cref{cor:segments_in_peripherals} implies that
  $F_\Omega(\psi(z)) = \{\psi(z)\}$.

  Then for any $x \in (u,v)$, $\rho(\gamma_n)x$ converges to
  $\psi(z)$, and we are done by
  \Cref{prop:attracting_spaces_in_faces}.
\end{proof}

\begin{proof}[Proof of \Cref{thm:convex_cocompact_boundary_dynamics}]
  Fix a $(d+1)$-dimensional real vector space $V$, let
  $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and let
  $\rho:\Gamma \to \PGL(V)$ be a representation such that
  $\rho(\Gamma)$ acts convex cocompactly on a properly convex domain
  $\Omega \subset \P(V)$. This implies (see \cite{iz2022structure})
  that each $H \in \mc{H}$ also acts convex cocompactly on $\Omega$.

  The first step in the proof is to define a $\rho(\Gamma)$-invariant
  subset $\hat{\Lambda} \subset \projflags(V) = \PGL(V)/P_{1,d}$ and a
  boundary extension $\hat{\phi}:\hat{\Lambda} \to \bgamh$. We
  use the map $\psi$ coming from
  \Cref{thm:convex_cocompact_boundary_embedding} to define an
  equivariant surjection $\phi:\Lambda \to \bgamh$, where the
  preimage of each parabolic point $p$ in $\bgamh$ is exactly
  $\Lambda_\Omega(H)$ for $H = \mr{Stab}_\Gamma(p)$.

  If we let $\Lambda^*$ be the full orbital limit set of the
  $\Gamma$-action on $\Omega^*$, we can similarly find an equivariant
  surjection $\phi^*:\Lambda^* \to \bgamh$, where
  ${\phi^*}^{-1}(z)$ is a single hyperplane if $z$ is a conical limit
  point, and the dual full orbital limit set of $\mr{Stab}_\Gamma(z)$
  if $z$ is a parabolic point.

  We consider the set
  \[
    \hat{\Lambda} = \{ (x, w) \in \projflags : x \in \Lambda, w \in
    \Lambda^*\}.
  \]

  Each element of $\Lambda^*$ is a supporting hyperplane of the domain
  $\Omega$. \Cref{cor:segments_in_peripherals} implies that for every
  point $(x, w)$ in $\hat{\Lambda}$, either:
  \begin{enumerate}
  \item $x = \phi^{-1}(z)$ and $w = {\phi^*}^{-1}(z)$ for a conical
    limit point $z \in \bgamh$, or
  \item $x \in \Lambda_H$ and $w \in \Lambda_H^*$ for a peripheral
    subgroup $H \in \mc{H}$.
  \end{enumerate}
  This allows us to combine $\phi$ and $\phi^*$ to get a well-defined
  equivariant surjection $\hat{\phi}:\hat{\Lambda} \to \bgamh$.

  The next step is to define the open subsets $C_z \subset \projflags$
  for each $z \in \bgamh$. If $z \in \bgamh$ is a conical
  limit point, we define the set $C_z$ by
  \[
    C_z = \{\nu \in \projflags : \nu \textrm{ is opposite to }
    \phi^{-1}(z)\}.
  \]

  Otherwise, if $z$ is a parabolic point, we consider the maximal
  domain $\maxdomain(H) \subset \P(V)$ for $H =
  \Stab_\Gamma(z)$. Dually, we can define
  $\Omega^*_{\mathrm{max}}(H) \subset \P(V^*)$, viewing
  $\Lambda_\Omega(H)$ as the dual full orbital limit set of $H^*$
  acting on $\Omega^*$.

  Then, we define
  \[
    C_z = \{(x, w) \in \projflags(V) : x \in \maxdomain(H), w \in
    \Omega^*_{\mathrm{max}}(H)\}.
  \]
  
  For every $z \in \bgamh$, $C_z$ is open, and
  \Cref{cor:segments_in_peripherals} implies that $C_z$ contains
  $\hat{\phi}^{-1}(z')$ for every $z' \ne z$ in $\dee
  \Gamma$.

  The last step is to check that the map $\hat{\phi}$ actually extends
  convergence dynamics, using the sets $C_z$. Because of
  \Cref{prop:conical_peripheral_implies_egf}, it suffices to verify
  that $\hat{\phi}$ has the right dynamical behavior for conical limit
  sequences and sequences in peripheral subgroups.

  Fix a conical limit point $z \in \bgamh$, and let
  $\gamma_n \in \Gamma$ be a sequence limiting conically to $z$, with
  $\gamma_n^{-1}$ limiting conically to some point $z_-$.

  \Cref{prop:boundary_embedding_conical_pts} implies that
  $\phi^{-1}(z)$ is the unique one-dimensional attracting subspace for
  $\rho(\gamma_n)$ in $\P(V)$, and $\phi^{-1}(z_-)$ is the unique
  one-dimensional attracting subspace for $\rho(\gamma_n^{-1})$ in
  $\P(V)$. Dually, $(\phi^*)^{-1}(z_-)$ is the unique one-dimensional
  attracting subspace for $\rho(\gamma_n^{-1})$ in $\P(V^*)$, and
  $(\phi^*)^{-1}(z)$ is the unique one-dimensional attracting subspace
  for $\rho(\gamma_n)$ in $\P(V^*)$. In particular, the sequences
  $\rho(\gamma_n^{\pm1})$ are $P_1$-divergent. Then
  \Cref{lem:klp_contraction_symmetry} ensures that $\rho(\gamma_n)$
  converges to $\hat{\phi}^{-1}(z)$ uniformly on compacts in
  $C_{z_-}$.

  On the other hand, if $z$ is a parabolic point and $\gamma_n$ is an
  infinite sequence in $\Stab_\Gamma(z)$,
  \Cref{prop:maxdomain_accumulation} implies that for any compact
  $K \subset C_z$, the set $\rho(\gamma_n) \cdot K$ eventually lies in
  any given neighborhood of $\phi^{-1}(z)$, as required.
\end{proof}


\subsection{Discreteness and finite kernel}
\label{sec:discrete_finite_kernel}

When $\rho:\Gamma \to G$ is EGF, the action of $\rho(\Gamma)$ on the
boundary set $\Lambda$ is by definition an extension of the
topological dynamical system $(\Gamma, \bgamh)$. When $\Gamma$ is
non-elementary, convergence dynamics imply that the homomorphism
$\Gamma \to \Homeo(\bgamh)$ has finite kernel and discrete image. So
the map $\Gamma \to \Homeo(\Lambda)$ must also have discrete image and
finite kernel, and therefore so does the representation
$\rho:\Gamma \to G$. The case where $\Gamma$ is elementary can be
verified directly.

\subsection{Shrinking the sets $C_z$}
Let $\rho:\Gamma \to G$ be an EGF representation with boundary map
$\phi:\Lambda \to \bgamh$. By assumption, we know there exists an
open subset $C_z \subset \flags$ for each $z \in \bgamh$,
satisfying the \emph{extended convergence dynamics} conditions
(\Cref{defn:extends_convergence_dynamics}). In general, there is not a
canonical choice for the set $C_z$. We are able to make some
assumptions about the properties of the $C_z$, however.

\begin{prop}
  \label{prop:point_cell_transversality}
  Let $\rho:\Gamma \to G$ be an EGF representation with boundary
  extension $\phi$. For any $z \in \bgamh$, we can choose the set
  $C_z$ to be a subset of
  \[
    \Opp(\phi^{-1}(z)) := \{\xi \in \flags : \xi \textrm{ is opposite
      to } \nu \textrm{ for every } \nu \in \phi^{-1}(z)\}.
  \]
\end{prop}
\begin{proof}
  Since $\phi^{-1}(z)$ is closed, $\Opp(\phi^{-1}(z))$ is an open subset
  of $\flags$. And, transversality of $\phi$ implies that
  $\Opp(\phi^{-1}(z))$ contains $\Lambda - \phi^{-1}(z)$. So the
  intersection $C_z \cap \Opp(\phi^{-1}(z))$ is open and nonempty,
  meaning we can replace $C_z$ with this intersection.
\end{proof}

\subsection{An equivalent characterization of EGF representations}

Let $\rho:\Gamma \to G$ be a representation of a relatively hyperbolic
group. It turns out than in order to check that $\rho$ is an EGF
representation, it suffices to look at the dynamics of the
$\rho(\Gamma)$-action along \emph{conical limit sequences} and
sequences lying entirely in peripheral subgroups of $\Gamma$.

\begin{restatable}{prop}{conicalPeripheralEGF}
  \label{prop:conical_peripheral_implies_egf}
  Let $\rho:\Gamma \to G$ be a representation of a relatively
  hyperbolic group, and let $\Lambda \subset \flags$ be a closed
  $\rho(\Gamma)$-invariant set, where $P \subset G$ is a symmetric
  parabolic subgroup. Suppose that $\phi:\Lambda \to \bgamh$ is a
  continuous surjective $\rho$-equivariant antipodal map.

  Then $\rho$ is an EGF representation if and only if for each
  $z \in \bgamh$, there exists $C_z \subset \flags$, with
  $C_z \subset \Opp(\phi^{-1}(z))$ and
  $\Lambda - \phi^{-1}(z) \subset C_z$, such that:
  \begin{enumerate}
  \item For any sequence $\gamma_n \in \Gamma$ limiting conically to
    $z$ (with $\gamma_n^{-1} \to z_-$), any compact
    $K \subset C_{z_-}$, and any neighborhood $U$ containing
    $\phi^{-1}(z)$, we have $\rho(\gamma_n) \cdot K \subset U$ for all
    sufficiently large $n$.
  \item For any parabolic point $p$, any compact $K \subset C_p$, and
    any open set $U$ containing $\phi^{-1}(p)$, for all but finitely
    many $\gamma \in \Gamma_p$, we have
    $\rho(\gamma) \cdot K \subset U$.
  \end{enumerate}
\end{restatable}

In practice, \Cref{prop:conical_peripheral_implies_egf} gives a
criterion for extended geometrical finiteness which is much easier to
check than the full definition, so we will use it throughout the
paper. However, the proof requires the technical machinery of
\emph{relative quasigeodesic automata}, so we defer it to
\Cref{sec:weak_egf}.

\subsection{Properties of $\Lambda$}

\begin{prop}
  \label{prop:boundary_contains_limset}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and let
  $\rho:\Gamma \to G$ be a representation which is EGF with respect to
  a symmetric parabolic $P$, with boundary extension
  $\phi:\Lambda \to \bgamh$. Then $\Lambda$ contains the $P$-limit set
  of $\rho(\Gamma)$.
\end{prop}
\begin{proof}
  Let $\xi \in \flags$ be a flag in the $P$-limit set of
  $\rho(\Gamma)$. Then there is a $P$-contracting sequence
  $\rho(\gamma_n)$ for $\gamma_n \in \Gamma$ and a flag
  $\xi_- \in G/P$ such that $\rho(\gamma_n)\eta$ converges to $\xi$
  for any $\eta$ in $\Opp(\xi_-)$. Up to subsequence
  $\gamma_n^{\pm 1}$ converges to $z_\pm \in \bgamh$, so for any flag
  $\eta \in C_{z_-}$, the sequence $\rho(\gamma_n)\eta$ subconverges
  to a point in $\phi^{-1}(z_+)$. But since $\Opp(\xi_-)$ is open and
  dense, for some $\eta \in C_{z_-}$ we have
  $\rho(\gamma_n)\eta \to \xi$ and hence $\xi \in \phi^{-1}(z_+)$.
\end{proof}

In particular, \Cref{prop:boundary_contains_limset} implies that the
EGF boundary set $\Lambda \subset \flags$ of an EGF representation
$\rho:\Gamma \to G$ must always contain the \emph{$P$-proximal limit
  set} of $\rho(\Gamma)$. (Recall that $g \in G$ is
\emph{$P$-proximal} if it has a unique attracting fixed point in
$\flags$; the \emph{$P$-proximal limit set} of a subgroup of $G$ is
the closure of the set of attracting fixed points of $P$-proximal
elements).

We will see that most of the power of EGF representations lies in the
fact that their associated boundary extensions
$\phi:\Lambda \to \bgamh$ do not have to be homeomorphisms (so
the Bowditch boundary of $\Gamma$ does not need to be equivariantly
embedded in any flag manifold). However, it turns out that it is
always possible to choose the boundary extension $\phi$ so that it has
a well-defined inverse on \emph{conical limit points} in
$\bgamh$. In fact, we can even get a somewhat precise description
of all the fibers of $\phi$. Concretely, we have the following:
\begin{prop}
  \label{prop:conical_pts_singletons}
  Let $\rho:\Gamma \to G$ be an EGF representation, with boundary
  extension $\phi:\Lambda \to \bgamh$. There is a
  $\rho(\Gamma)$-invariant closed subset $\Lambda' \subset \flags$ and
  a $\rho$-equivariant map $\phi':\Lambda' \to \bgamh$
  such that:
  \begin{enumerate}
  \item $\phi':\Lambda' \to \bgamh$ is also a boundary
    extension for $\rho$,
  \item for every $z \in \conbdry(\Gamma, \mc{H})$, $\phi'^{-1}(z)$ is
    a singleton, and
  \item for every $p \in \parbdry(\Gamma, \mc{H})$, $\phi'^{-1}(p)$ is
    the closure of the set of all accumulation points of orbits
    $\gamma_n \cdot x$ for $\gamma_n$ a sequence of distinct elements
    in $\Gamma_p$ and $x \in C_p$.
  \end{enumerate}
\end{prop}

We will prove \Cref{prop:conical_pts_singletons} at the end of
\Cref{sec:relative_stability}, where it will follow as a consequence
of the proof of the relative stability theorem for EGF representations
(\Cref{thm:cusp_stable_stability})---see
\Cref{rem:conical_pts_singletons}.

We will rely on both \Cref{prop:conical_peripheral_implies_egf} and
\Cref{prop:conical_pts_singletons} to prove the rest of the results in
this section (which are not needed anywhere else in
\Cref{part:egfreps} of this paper).

\subsection{Relatively asymptotically embedded representations}

EGF representations give a strict generalization of the
\emph{relatively asymptotically embedded} representations of Kapovich
and Leeb. We recall the definition here.

\begin{definition}[\cite{kl2018relativizing}, Definition 7.1]
  Let $\Gamma$ be a subgroup of $G$ and suppose $(\Gamma, \mc{H})$ is
  a relatively hyperbolic pair. Let $P_\taumod \subset G$ be a
  symmetric parabolic subgroup.

  The subgroup $\Gamma$ is \emph{relatively $\taumod$-asymptotically
    embedded} if it is $P_\taumod$-divergent, and there is a
  $\Gamma$-equivariant antipodal embedding $\bgamh \to \flags$ whose
  image $\Lambda_\taumod$ is the $P_\taumod$-limit set of $\Gamma$.
\end{definition}

Here, we say an embedding $\psi:\bgamh \to \flags$ is
\emph{antipodal} if for every distinct $\xi_1, \xi_2$ in
$\bgamh$, $\psi(\xi_1)$ and $\psi(\xi_2)$ are opposite
flags.

\begin{remark}
  When $\Gamma$ is a \emph{hyperbolic} group (and the collection of
  peripheral subgroups $\mc{H}$ is empty), then the Bowditch boundary
  $\bgamh$ is identified with the Gromov boundary
  $\dee \Gamma$. In this case, a relatively asymptotically embedded
  representation is just called an \emph{asymptotically embedded}
  representation. $\taumod$-asymptotically embedded representations
  are exactly the same as $P_\taumod$-Anosov representations (see
  \cite{klp2017anosov}, Theorem 1.1).

  In general, it is possible to define $P$-Anosov representations for
  a \emph{non-symmetric} parabolic subgroup $P$. However, there is no
  loss of generality in assuming that $P$ is symmetric: a
  representation $\rho:\Gamma \to G$ is $P$-Anosov if and only if it
  is $P'$-Anosov for a symmetric parabolic subgroup $P' \subset G$
  depending only on $P$.
\end{remark}

\begin{prop}
  \label{prop:homeomorphism_implies_asymptotically_embedded}
  Let $\rho:\Gamma \to G$ be an EGF representation with respect to
  $P_\taumod$, and suppose that the boundary extension
  $\phi:\Lambda \to \bgamh$ is a homeomorphism. Then:
  \begin{enumerate}
  \item $\rho(\Gamma)$ is $P_\taumod$-divergent, and $\Lambda$ is the
    $P_\taumod$-limit set of $\rho(\Gamma)$.
    
  \item The sets $C_z$ for $z \in \bgamh$ can be taken to be
    \[
      \Opp(\phi^{-1}(z)) = \{\nu \in \flags : \nu \textrm{ is opposite to
      } \phi^{-1}(z)\}.
    \]
  \end{enumerate}
\end{prop}
\begin{proof}
  (1). Let $\gamma_n$ be any infinite sequence of elements in
  $\Gamma$. After extracting a subsequence, we have
  $\gamma_n^{\pm1} \to z_\pm$, and since $\phi$ is a homeomorphism,
  $\rho(\gamma_n)$ converges to the point $\phi^{-1}(z_+)$ uniformly
  on compacts in the open set $C_{z_-}$. Then
  \Cref{prop:local_contracting_implies_global_contracting} implies
  that $\rho(\gamma_n)$ is $P_\taumod$-divergent, with unique
  $P_\taumod$-limit point $\phi^{-1}(z_+) \in \Lambda$.

  (2). The fact that $\phi$ is antipodal is exactly the statement that
  the sets $\Opp(\phi^{-1}(z))$ contain $\Lambda - \phi^{-1}(z)$ for
  every $z \in \bgamh$, so we just need to see that the
  appropriate dynamics hold for these sets. Let $\gamma_n$ be an
  infinite sequence in $\Gamma$ with $\gamma_n^{\pm 1} \to z_{\pm}$
  for $z_\pm \in \bgamh$.

  We know that for open subsets $U_\pm \subset \flags$, we have
  $\rho(\gamma_n) \cdot U_+ \to \phi^{-1}(z_+)$ and
  $\rho(\gamma_n^{-1}) U_- \to \phi^{-1}(z_-)$, uniformly on
  compacts. \Cref{prop:local_contracting_implies_global_contracting}
  implies that $\rho(\gamma_n)$ and $\rho(\gamma_n^{-1})$ are both
  $P$-divergent with unique $P$-limit points $\phi^{-1}(z_+)$,
  $\phi^{-1}(z_-)$. So in fact by \Cref{lem:klp_contraction_symmetry}
  $\rho(\gamma_n)$ converges to $\phi^{-1}(z_+)$ uniformly on compacts
  in $\Opp(\phi^{-1}(z_-))$.
\end{proof}

Using the previous proposition, we can see the relationship between
relatively asymptotically embedded representations and EGF
(\Cref{thm:bdry_dynamics_homeo_equiv_rel_asymp_embedded}). We restate
the theorem here for convenience:

\relAsympBdryDynamics*

\begin{proof}
  \Cref{prop:homeomorphism_implies_asymptotically_embedded} ensures
  that if $\rho$ is an EGF representation, and the boundary extension
  $\phi$ is a homeomorphism, then $\rho$ is $P_\taumod$-divergent and
  $\phi^{-1}$ is an antipodal embedding whose image is the
  $P_\taumod$-limit set.

  On the other hand, if $\rho$ is relatively $\taumod$-asymptotically
  embedded, with boundary embedding
  $\psi:\bgamh \to \Lambda$, for each
  $z \in \bgamh$, we can take
  \[
    C_z = \Opp(\psi(z)).
  \]
  Antipodality means that $C_z$ contains $\Lambda - \psi(z)$, and
  $P_\taumod$-divergence and \Cref{lem:klp_contraction_symmetry} imply
  that $\rho(\Gamma)$ has the appropriate convergence dynamics.
\end{proof}

\subsection{Relativization}

We now turn to the situation where we have an EGF representation of a
\emph{hyperbolic} group $\Gamma$ with a \emph{nonempty} collection of
peripheral subgroups. That is, for some invariant set
$\Lambda \subset \flags$, we have an EGF boundary extension
$\phi:\Lambda \to \bgamh$, where $\bgamh$
is the Bowditch boundary of $\Gamma$ with peripheral structure
$\mc{H}$.

We want to prove \Cref{thm:changing_peripheral_structure}, which says
that in this situation, if $\rho$ restricts to a $P$-Anosov
representation on each $H \in \mc{H}$, then $\rho$ is a $P$-Anosov
representation of $\Gamma$. For the rest of this section, we assume
that $\Gamma$ is a hyperbolic group, and $\mc{H}$ is a collection of
subgroups of $\Gamma$ so that the pair $(\Gamma, \mc{H})$ is
relatively hyperbolic. We let $\rho:\Gamma \to G$ be an EGF
representation for the pair $(\Gamma, \mc{H})$ with respect to a
symmetric parabolic subgroup $P \subset G$, and we assume that for
each $H \in \mc{H}$, $\rho|_H:H \to G$ is $P$-Anosov, with Anosov
limit map $\psi_H:\dee H \to \flags$.

The main step in the proof is to observe that it is always possible to
choose the boundary extension $\phi:\Lambda \to \bgamh$
so that $\Lambda$ is equivariantly homeomorphic to the Gromov boundary
of $\Gamma$ (which we here denote $\dee \Gamma$).

Whenever $\Gamma$ is a hyperbolic group and $\mc{H}$ is a collection
of subgroups so that $(\Gamma, \mc{H})$ is a relatively hyperbolic
pair, there is an explicit description of the Bowditch boundary
$\bgamh$ in terms of the Gromov boundary $\dee \Gamma$
of $\Gamma$---see \cite{gerasimov2012floyd}, \cite{gp2013quasi}, or
\cite{tran2013relations}. Specifically, we can say:
\begin{prop}
  \label{prop:bb_quotient_description}
  There is an equivariant surjective continuous map
  $\phi_\Gamma:\dee \Gamma \to \bgamh$ such that for each conical
  limit point $z$ in $\bgamh$, $\phi_\Gamma^{-1}(z)$ is a singleton,
  and for each parabolic point $p \in \bgamh$ with
  $H = \Stab_\Gamma(p)$, $\phi_\Gamma^{-1}(p)$ is an embedded copy of
  $\dee H$ in $\dee \Gamma$.
\end{prop}

In our situation, we can see that the boundary extension
$\phi:\Lambda \to \bgamh$ satisfies similar properties.




\begin{lem}
  \label{lem:parabolic_preimages_anosov_images}
  There is a closed $\rho(\Gamma)$-invariant subset
  $\Lambda' \subset \flags$ and an EGF boundary extension
  $\phi':\Lambda' \to \bgamh$ such that:
  \begin{enumerate}
  \item For each conical limit point $z \in \bgamh$,
    $\phi'^{-1}(z)$ is a singleton.
  \item For each parabolic point $p \in \bgamh$, with
    $H = \Stab_\Gamma(p)$, we have $\phi'^{-1}(p) = \psi_H(\dee H)$.
  \end{enumerate}
\end{lem}

\begin{proof}
  We choose $\Lambda'$ as in \Cref{prop:conical_pts_singletons}. The
  only thing we need to check is that for $H = \Stab_\Gamma(p)$, the
  set $\psi_H(\dee H)$ is exactly the closure of the set of
  accumulation points of $\rho(H)$-orbits in $C_p$. But since we may
  assume $C_p$ is contained in $\Opp(\psi_H(\dee H))$, this follows
  immediately from the fact that $\rho(H)$ is $P$-divergent and the
  closed set $\psi_H(\dee H)$ is the $P$-limit set of $\rho(H)$.   
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
\end{proof}

Next we need a lemma which will allow us to characterize the Gromov
boundary of $\Gamma$ as an extension of the Bowditch boundary
$\bgamh$. First recall that if $\Gamma$ acts as a
convergence group on a space $Z$, the \emph{limit set} of $\Gamma$ is
the set of points $z \in Z$ such that for some $y \in Z$ and some
sequence $\gamma_n \in \Gamma$, we have
\[
  \gamma_n |_{Z - \{y\}} \to z
\]
uniformly on compacts.

\begin{lem}
  \label{lem:sequence_convergence_agrees}
  Let $\Gamma$ act on compact metrizable spaces $X$ and $Y$, and let
  $\phi_X:X \to \bgamh$, $\phi_Y:Y \to \bgamh$ be continuous
  equivariant surjective maps such that for every conical limit point
  $z \in \bgamh$, $\phi_X^{-1}(z)$ and $\phi_Y^{-1}(z)$ are both
  singletons, and for every parabolic point $p \in \bgamh$,
  $H = \Stab_\Gamma(p)$ acts as a convergence group on $X$ and $Y$,
  with limit sets $\phi_X^{-1}(p)$, $\phi_Y^{-1}(p)$ equivariantly
  homeomorphic to $\dee H$.

  Then for any sequences $z_n, z_n' \in \conbdry(\Gamma, \mc{H})$, we
  have
  \[
    \lim_{n \to \infty} \phi_X^{-1}(z_n) = \lim_{n \to \infty}\phi_X^{-1}(z_n')
  \]
  if and only if
  \[
    \lim_{n \to \infty} \phi_Y^{-1}(z_n) = \lim_{n \to
      \infty}\phi_Y^{-1}(z_n').
  \]
\end{lem}
\begin{proof}
  We proceed by contradiction, and suppose that for a pair of
  sequences $z_n, z_n' \in \conbdry(\Gamma, \mc{H})$, we have
  \[
    \lim_{n \to \infty} \phi_X^{-1}(z_n) = \lim_{n \to
      \infty}\phi_X^{-1}(z_n') = x,
  \]
  but
  \[
    \lim_{n \to \infty}\phi_Y^{-1}(z_n) \ne \lim_{n \to
    \infty}\phi_Y^{-1}(z_n').
  \]
  After taking a subsequence we may assume $z_n$ converges to
  $z \in \bgamh$, and that $y_n = \phi_Y^{-1}(z_n)$
  converges to $y$ and $y_n' = \phi_Y^{-1}(z_n')$ converges to $y'$
  for $y \ne y'$. By continuity, we have
  \[
    \phi_Y(y) = \phi_Y(y') = \phi_X(x) = z.
  \]

  Since $\phi_X$ and $\phi_Y$ are bijective on
  $\phi_X^{-1}(\conbdry(\Gamma, \mc{H}))$ and
  $\phi_Y^{-1}(\conbdry(\Gamma, \mc{H}))$ respectively, we must have
  $z = p$ for a parabolic point $p \in \parbdry(\Gamma, \mc{H})$. Let
  $H = \Stab_\Gamma(p)$.

  Since $p$ is a bounded parabolic point, we can find sequences of
  group elements $h_n, h_n' \in H$ so that for a fixed compact subset
  $K \subset \bgamh - \{p\}$, we have
  \begin{equation}
    \label{eq:bounded_parabolic}
    h_nz_n \in K, \quad h_n'z_n' \in K. 
  \end{equation}
  This implies that no subsequence of $h_n y_n$ or $h_n'y_n'$
  converges to a point in $\phi_Y^{-1}(p)$.

  Then, since $H$ acts as a convergence group on $Y$ with limit set
  $\phi_Y^{-1}(p)$, up to subsequence there are points
  $u, u' \in \phi_Y^{-1}(p)$ so that $h_n$ converges to a point in
  $\phi_Y^{-1}(p)$ uniformly on compacts in $Y - \{u\}$, and $h_n'$
  converges to a point in $\phi_Y^{-1}(p)$ uniformly on compacts in
  $Y - \{u'\}$. So, we must have $u = y$ and $u' = y'$.

  This means that the sequences $h_n^{-1}$ and $h_n'^{-1}$ have
  distinct limits in the compactification
  $\overline{H} = H \sqcup \dee H$. So, there are distinct points
  $v, v' \in \phi_X^{-1}(p)$ so that (again up to subsequence) $h_n$
  converges to a point in $\phi_X^{-1}(p)$ uniformly on compacts in
  $X - \{v\}$, and $h_n'$ converges to a point in $\phi_X^{-1}(p)$
  uniformly on compacts in $X - \{v'\}$. Without loss of generality,
  we can assume $x \ne v$.

  But then $\phi_X^{-1}(z_n)$ lies in a compact subset of $X - \{v\}$,
  so $h_n\phi_X^{-1}(z_n)$ converges to a point in $\phi_X^{-1}(p)$
  and $h_nz_n$ converges to $p$. But this contradicts
  (\ref{eq:bounded_parabolic}) above.
\end{proof}

\begin{prop}
  \label{prop:gromov_boundary_homeo}
  If the set $\Lambda$ satisfies the conclusions of
  \Cref{lem:parabolic_preimages_anosov_images}, then $\Lambda$ is
  equivariantly homeomorphic to the Gromov boundary of $\Gamma$.
\end{prop}
\begin{proof}  
  Let $\phi_\Gamma : \dee \Gamma \to \bgamh$ denote the
  quotient map identifying the limit set of each $H \in \mc{H}$ to the
  parabolic point $p$ with $H = \Stab_\Gamma(p)$. For each conical
  limit point $z \in \bgamh$, the fiber
  $\phi_\Gamma^{-1}(z)$ is a singleton. So, there is an equivariant
  bijection $f$ from $\phi_\Gamma^{-1}(\conbdry(\Gamma, \mc{H}))$ to
  $\phi^{-1}(\conbdry(\Gamma, \mc{H}))$.

  Moreover, since $\phi_\Gamma^{-1}(\conbdry(\Gamma, \mc{H}))$ is
  $\Gamma$-invariant, and the action of $\Gamma$ on its Gromov
  boundary $\dee \Gamma$ is minimal,
  $\phi_\Gamma^{-1}(\conbdry(\Gamma, \mc{H}))$ is dense in
  $\dee \Gamma$. We claim that $f$ extends to a continuous injective
  map $\dee \Gamma \to \Lambda$ by defining $f(x) = \lim f(x_n)$ for
  any sequence $x_n \to x$.

  To see this, we can apply \Cref{lem:sequence_convergence_agrees},
  taking $\dee \Gamma = X$ and $\Lambda = Y$. We know that $\Gamma$
  always acts on its own Gromov boundary as a convergence group (so in
  particular each $H \in \mc{H}$ acts on $\dee \Gamma$ as a
  convergence group with limit set $\dee H$). And, since $\rho$
  restricts to a $P$-Anosov representation on each $H \in \mc{H}$, for
  any infinite sequence $h_n \in H$, up to subsequence there are
  $u, u_- \in \psi_H(\dee H)$ so that $\rho(h_n)$ converges to $u$
  uniformly on compacts in $\Opp(u_-)$. Antipodality of $\phi$ implies
  that $\rho(h_n)$ converges to $u$ uniformly on compacts in
  $\Lambda - \psi_H(\dee H)$. The other hypotheses of
  \Cref{lem:sequence_convergence_agrees} follow from
  \Cref{prop:bb_quotient_description} and
  \Cref{lem:parabolic_preimages_anosov_images}.

  We still need to check that $f$ is actually surjective. We know that
  $f$ restricts to a bijection on
  $\phi_\Gamma^{-1}(\conbdry(\Gamma, \mc{H}))$, and that $f$ takes
  $\phi_\Gamma^{-1}(p)$ to $\phi^{-1}(p)$ for each parabolic point $p$
  in $\bgamh$. So we just need to check that for every
  $H \in \mc{H}$, $f$ restricts to a surjective map
  $\dee H \to \psi_H(\dee H)$. If $H$ is non-elementary, this must be
  the case because the action of $H$ on $\dee H$ is minimal and $f$
  maps $\dee H$ into $\psi_H(\dee H)$ as an invariant closed
  subset. Otherwise, $H$ is virtually cyclic and $\dee H$,
  $\psi_H(\dee H)$ both contain exactly two points. Then injectivity
  of $f$ implies surjectivity.

  So we conclude that there is a continuous bijection
  $f:\dee \Gamma \to \Lambda$, and since $\dee \Gamma$ is compact and
  $\Lambda$ is metrizable, $f$ is a homeomorphism.
\end{proof}

We let $f:\Lambda \to \dee \Gamma$ denote the equivariant
homeomorphism from \Cref{prop:gromov_boundary_homeo}. The final step
in the proof of \Cref{thm:changing_peripheral_structure} is the
following:
\begin{prop}
  \label{prop:egf_anosov_extends_convergence}
  The equivariant homeomorphism $f:\Lambda \to \dee \Gamma$ extends
  the convergence dynamics of $\Gamma$ on its Gromov boundary
  $\dee \Gamma$.
\end{prop}
\begin{proof}
  By \Cref{prop:conical_peripheral_implies_egf}, we just need to show
  that if $\gamma_n \in \Gamma$ is a conical limit sequence with
  $\gamma_n^{\pm 1} \to z_{\pm}$ for $z_\pm \in \dee \Gamma$, then
  $\rho(\gamma_n)$ converges to $\phi^{-1}(z_+)$ uniformly on compacts
  in $\Opp(\phi^{-1}(z_-))$.

  We consider two cases:
  \begin{aside}{Case 1: $\phi \circ f(z_+)$ is a parabolic point $p$
      in $\bgamh$}
    In this case, $\gamma_n$ lies along a quasigeodesic ray in
    $\Gamma$ limiting to some $z_+ \in \dee H$. This means that for a
    bounded sequence $b_n \in \Gamma$, we have $\gamma_nb_n \in
    H$. Since $\rho$ restricts to a $P$-Anosov representation on $H$,
    this means that $\rho(\gamma_nb_n)$ is $P$-divergent with unique
    $P$-limit point $\psi_H(z_+)$. But then $\rho(\gamma_n)$ is also
    $P$-divergent with unique $P$-limit point $\psi_H(z_+)$.

    Since $p$ is a parabolic point (in particular, \emph{not} a
    conical limit point), and $\gamma_n$ limits to $p$, the sequence
    $\gamma_n^{-1}$ must also limit to $p$. So we apply the same
    reasoning to see that $\rho(\gamma_n^{-1})$ is also $P$-divergent
    with unique $P$-limit point $\psi_H(z_-)$. Then we are done by
    \Cref{lem:klp_contraction_symmetry}. 
  \end{aside}
  \begin{aside}{Case 2: $\phi \circ f(z_+)$ is a conical limit point
      in $\bgamh$}
    In this case, subsequences of both $\gamma_n$ and $\gamma_n^{-1}$
    are conical limit sequences for the action of $\Gamma$ on
    $\bgamh$. So, $\phi \circ f(z_-)$ is also a conical
    limit point in $\bgamh$. So the desired result
    follows directly from the fact that $\rho$ is an EGF
    representation.
  \end{aside}
\end{proof}

\begin{proof}[Proof of \Cref{thm:changing_peripheral_structure}]
  Let $\Gamma$ be hyperbolic, let $\mc{H}$ be a collection of
  subgroups such that $(\Gamma, \mc{H})$ is a relatively hyperbolic
  pair, and let $\rho:\Gamma \to G$ be an EGF representation with
  respect to $P$, for the peripheral structure $\mc{H}$.

  Suppose that $\rho$ restricts to a $P$-Anosov representation on each
  $H \in \mc{H}$. \Cref{prop:egf_anosov_extends_convergence} implies
  that $\rho$ is \emph{also} an EGF representation of $\Gamma$ for its
  \emph{empty} peripheral structure, whose boundary extension can be
  chosen to be a homeomorphism. Then
  \Cref{thm:bdry_dynamics_homeo_equiv_rel_asymp_embedded} says that
  $\rho$ is relatively asymptotically embedded (again for the empty
  peripheral structure on $\Gamma$). Then we apply
  \Cref{cor:hyperbolic_egf_anosov} to complete the proof.
\end{proof}


\subsection{Construction of the relative automaton}

We will construct the relative automaton $\mc{G}$ satisfying the
conditions of \Cref{prop:convergence_dynamics_implies_automaton} by
choosing a suitable open covering of $\bgamh$, and then using
compactness to take a finite subcover. The subsets of this subcover
will be the vertices of $\mc{G}$.

We choose constants $\eps > 0$, $\delta > 0$ so that
$\eps < \eps_{\mr{con}}$, $\delta < \delta_{\mr{con}}$ (where
$\eps_{\mr{con}}$, $\delta_{\mr{con}}$ are the constants coming from
\Cref{lem:conical_limit_neighborhoods}) and $\eps < \eps_p$,
$\delta < \delta_p$ for each $p \in \Pi$ (where $\eps_p, \delta_p$ are
the constants coming from \Cref{lem:large_parabolic_neighborhood}).

Then:
\begin{itemize}
\item For each $z \in \conbdry\Gamma$, we define $W_z$, $V_z$,
  $\gamma_z$ as in \Cref{lem:conical_limit_neighborhoods}, with
  parameters $\eps$, $\delta$.
\item For each $q \in \parbdry \Gamma$, we define
  $V_q, W_q, \hat{V}_q, \hat{W}_q$, and $T_q$ as in
  \Cref{lem:large_parabolic_neighborhood}, again with parameters
  $\eps$, $\delta$.
\end{itemize}
The collections of sets $\{V_z : z \in \conbdry \Gamma\}$ and
$\{V_q : q \in \parbdry\Gamma\}$ together give an open covering of
the Bowditch boundary $\bgamh$. So we choose a finite subcover
$\mc{V}$, which we can write as
\[
  \mc{V} = \{V_a : a \in A\}
\]
where $A$ is a finite subset of $\bgamh$. We can in
particular ensure that $A$ contains the finite set $\Pi$.

We identify the vertices of our $\Gamma$-graph $\mc{G}$ with $A$. For
each $a \in A$, the set $T_a$ is either $\{\gamma_a\}$ (if $a$ is a
conical limit point) or $T_q$ (if $a = q$ for a parabolic point
$q$). Then, for each $a \in A$, we define the open sets $U_a$ by
\[
  U_a = N_M(\phi^{-1}(a), \eps).
\]

The edges of the $\Gamma$-graph $\mc{G}$ are defined as follows:
\begin{itemize}
\item For $a, b \in A$ with $a \in \conbdry\Gamma$, there is an edge
  from $a$ to $b$ if $(\gamma_a^{-1} \cdot V_a) \cap V_b$ is nonempty.
\item If $a, b \in A$ with $a \in \parbdry\Gamma$, there is an edge
  from $a$ to $b$ if $\hat{V}_a \cap V_b$ is nonempty.
\end{itemize}

Since $\mc{V}$ is an open covering of $\bgamh$, and the sets
$\hat{V}_a$ and $\gamma_a^{-1}V_a$ are nonempty, every vertex of
$\mc{G}$ has at least one outgoing edge. Moreover, for any parabolic
point $a$, the set $\hat{V}_a$ depends only on the orbit of $a$ in
$\bgamh$, so $\mc{G}$ must satisfy condition (2) in
\Cref{prop:convergence_dynamics_implies_automaton}.
\begin{prop}
  \label{lem:small_nbhds_in_lg_nbhds}
  For each $a \in A$, we have
  \[
    \phi^{-1}(W_a) \subset U_a.
  \]
\end{prop}
\begin{proof}
  When $a$ is not a parabolic vertex, Part (3) of
  \Cref{lem:conical_limit_neighborhoods} implies:
  \[
    \phi^{-1}(W_a) = \gamma_a\gamma_a^{-1}\phi^{-1}(W_a) \subset
    \gamma_aN(\gamma_a^{-1}\phi^{-1}(W_a), 2\eps) \subset
    N_M(\phi^{-1}(a), \eps) = U_a.
  \]
  When $a$ is a parabolic vertex, then the claim follows directly from
  Part (1) of \Cref{lem:large_parabolic_neighborhood}.
\end{proof}

Next we verify:

\begin{prop}
  \label{prop:edge_inclusions}
  The collection of sets $\{W_v\}$ and $\{U_v\}$ are both
  $\mc{G}$-compatible systems of open sets for the $\Gamma$-graph
  $\mc{G}$.
\end{prop}
\begin{proof}
  First fix an edge $(a, b)$ with $a \in \conbdry \Gamma$. Since
  $(\gamma_a^{-1}V_a) \cap V_b$ is nonempty, part
  \ref{item:conical_nbhds_nested} of
  \Cref{lem:conical_limit_neighborhoods} implies that
  $\gamma_a^{-1} \cdot W_a$ contains the $\delta$-neighborhood of some
  point $z \in V_b$. Since $\mr{diam}(W_b) < \delta$ and
  $V_b \subset W_b$, we can find a small $\eps' > 0$ so that
  $\gamma_aN_{\dee\Gamma}(W_b, \eps') \subset W_a$.

  In particular, $\gamma_a^{-1} \cdot W_a$ contains $b$, which means
  that $N_M(\gamma_a^{-1}\phi^{-1}(W_a), 2\eps)$ contains
  $N_M(\phi^{-1}(b), 2\eps)$, which contains $N_M(U_b, \eps)$. Then,
  part \ref{item:conical_eps_nbhds_nested} of
  \Cref{lem:conical_limit_neighborhoods} implies that
  $\gamma_a \cdot N_M(U_b, \eps)$ is contained in
  $N_M(\phi^{-1}(a), \eps) = U_a$.

  Next fix an edge $(q, b)$ with $q \in \parbdry \Gamma$. From part
  \ref{item:parabolic_nbhds_nested} of
  \Cref{lem:large_parabolic_neighborhood}, we know that $\hat{W}_q$
  contains the $\delta$-neighborhood of a point
  $z \in \hat{V}_q \cap V_b$. Since $\mr{diam}(W_b) < \delta$ and
  $V_b \subset W_b$, this means that $\hat{W}_q$ contains an
  $\eps'$-neighborhood of $W_b$ for some small $\eps' > 0$. So part
  \ref{item:parabolic_eps_nbhds_nested} of
  \Cref{lem:large_parabolic_neighborhood} implies that for any
  $\gamma \in T_q$, we have $\gamma \cdot N(W_b, \eps') \subset W_q$.

  In particular $\hat{W}_q$ contains $b$, so
  $N_M(\phi^{-1}(\hat{W}_q), 2\eps)$ contains
  $N_M(\phi^{-1}(b), 2\eps)$, which contains $N_M(U_b, \eps)$. Then,
  part \ref{item:parabolic_eps_nbhds_nested} of
  \Cref{lem:large_parabolic_neighborhood} implies that
  \[
    \gamma N_M(U_b, \eps) \subset N_M(\phi^{-1}(q), \eps) = U_q
  \]
  for any $\gamma \in T_q$.
\end{proof}

We observe:
\begin{prop}
  The $\mc{G}$-compatible systems of open subsets $\{U_v\}$ and
  $\{W_v\}$ satisfy (1) - (3) in
  \Cref{prop:convergence_dynamics_implies_automaton}.
\end{prop}
\begin{proof}
  Part (1) follows directly from \Cref{lem:small_nbhds_in_lg_nbhds},
  and the fact that we defined each $U_a$ to be the
  $\eps$-neighborhood of $\phi^{-1}(a)$. Part (2) is true by the
  construction of the open covering $\mc{V}$ and the graph
  $\mc{G}$. Part (3) follows by construction and from part
  \ref{item:parabolic_in_attracting} of
  \Cref{lem:large_parabolic_neighborhood}.
\end{proof}

To finish the proof of
\Cref{prop:convergence_dynamics_implies_automaton}, we now just need
to show:
\begin{prop}
  \label{prop:graph_is_automaton}
  $\mc{G}$ is a relative quasigeodesic automaton for the pair
  $(\Gamma, \mc{H})$.
\end{prop}
\begin{proof}
  We apply \Cref{prop:relative_geodesic_automaton_from_bb}, using the
  $\mc{G}$-compatible system $\{W_a\}$ and the sets $\{V_a\}$ we
  defined in the construction of $\mc{G}$.

  The first three conditions of
  \Cref{prop:relative_geodesic_automaton_from_bb} are satisfied by
  construction. To see that conditions \ref{item:conical_edge_exists}
  and \ref{item:parabolic_edge_exists} hold, first observe that if
  $z \in V_a$ for a non-parabolic vertex $a$, then
  $\gamma_a^{-1} \cdot z$ lies in some $V_b$ and $(a,b)$ is an edge in
  $\mc{G}$. And if $z \in V_a - \{p_a\}$ for a parabolic vertex $a$,
  then part (\ref{item:parabolic_backward_edge}) of
  \Cref{lem:large_parabolic_neighborhood} says that there is some
  $\gamma \in T_a$ such that $\gamma^{-1} \cdot z \in \hat{V}_a$. If
  $V_b$ contains $\gamma^{-1} \cdot z$, the edge $(a,b)$ must be in
  $\mc{G}$.

  It only remains to check that $\mc{G}$ is a divergent
  $\Gamma$-graph. Let $\{\alpha_n\}$ be an infinite $\mc{G}$-path,
  following a vertex path $\{v_n\}$. The $\mc{G}$-compatibility
  condition implies that $\gamma_n \overline{U}_{v_{n+1}}$ is a subset
  of $\gamma_{n-1} U_{v_n}$ for every $n$. Since $M$ is connected and
  each $U_v$ is a proper subset of $M$, this inclusion must be
  proper. This implies that in the sequence $\gamma_n$, no element can
  appear more than $\#A$ times and therefore $\gamma_n$ is divergent.
\end{proof}

\begin{remark}
  This last step is the only part of the proof of
  \Cref{prop:convergence_dynamics_implies_automaton} which uses the
  connectedness of $M$. This hypothesis is likely unnecessary, but
  omitting it would involve introducing additional technicalities in
  the construction of the sets $V_a$, $W_a$---and as stated, the
  proposition is strong enough for our purposes.

  Note that with this hypothesis removed,
  \Cref{prop:convergence_dynamics_implies_automaton} would imply that
  any non-elementary relatively hyperbolic group has a relative
  quasigeodesic automaton (by taking $M = \bgamh$). As stated, the
  proposition only shows that such an automaton exists when $\bgamh$
  is connected.
\end{remark}

We conclude this section by observing that one can slightly refine the
construction in \Cref{prop:convergence_dynamics_implies_automaton} as
follows:
\begin{prop}
  \label{prop:compact_bowditch_nbhd}
  Fix a compact subset $Z$ of the Bowditch boundary $\bgamh$. Then,
  for any open set $U \subset M$ containing $\phi^{-1}(Z)$, there is a
  relative quasigeodesic automaton $\mc{G}$ and a pair of
  $\mc{G}$-compatible systems of open sets $\{U_a\}$, $\{W_a\}$ as in
  \Cref{prop:convergence_dynamics_implies_automaton}, additionally
  satisfying the following: any $z \in Z$ is the limit of a
  $\mc{G}$-path $\{\alpha_n\}$ (with corresponding vertex path
  $\{v_n\}$) such that $U_{v_1} \subset U$.
\end{prop}
\begin{proof}
  We choose $\eps > 0$ so that $U$ contains $N_M(\phi^{-1}(Z),
  \eps)$. We then construct our relative quasigeodesic automaton
  $\mc{G}$ as in the proof of
  \Cref{prop:convergence_dynamics_implies_automaton}, but we also
  choose a finite subset $A_Z \subset Z$ so that the sets $V_a$ for
  $a \in A_Z$ give a finite open covering of $Z$. We can ensure that
  the vertex set $A$ of $\mc{G}$ contains $A_Z$.
  
  Then, for any $z \in Z$, by the construction in
  \Cref{prop:relative_geodesic_automaton_from_bb}, we can find a
  $\mc{G}$-path limiting to $z$ whose first vertex is some
  $a \in A_Z$. The corresponding open set for this vertex is
  $U_a = N_M(\phi^{-1}(a), \eps) \subset U$.
\end{proof}


\subsection{Contracting paths in $\Gamma$-graphs}

\begin{definition}
  Let $\Gamma$ be a discrete subgroup of $G$, let $\mc{G}$ be a
  $\Gamma$-graph, and let $\{U_v\}_{v \in V(\mc{G})}$ be a
  $\mc{G}$-compatible system of open subsets of $\pflags$. We say that
  a $\mc{G}$-path $\{\alpha_n\}_{n \in \N}$ is \emph{contracting} if
  the decreasing intersection
  \begin{equation}
    \label{eq:contracting_intersection}
    \bigcap_{n=1}^\infty \alpha_1 \cdots \alpha_n \cdot U_{v_{n+1}}
  \end{equation}
  is a singleton in $\pflags$.
\end{definition}

\begin{definition}
  We say that an open set $\Omega \subset \pflags$ is a \emph{proper
    domain} if the closure of $\Omega$ lies in an affine chart
  $\Opp(\xi) \subset \pflags$ for some $\xi \in \oppflags$.
\end{definition}

Here is the main result in this section:
\begin{prop}
  \label{prop:compatible_system_in_flags_contracting}
  Let $\mc{G}$ be a $\Gamma$-graph for $(\Gamma, \mc{H})$, and let
  $\{U_v\}_{v \in V(\mc{G})}$ be $\mc{G}$-compatible system of open
  subsets of $\pflags$.

  If the set $U_v$ is a proper domain for each vertex $v$ of the
  automaton, then every infinite $\mc{G}$-path is contracting.
\end{prop}


\subsection{A metric property for bounded domains in flag manifolds}

To prove \Cref{prop:compatible_system_in_flags_contracting}, we
consider a metric $C_\Omega$ defined by Zimmer \cite{zimmer2018proper}
on any proper domain $\Omega \subset \pflags$. $C_\Omega$ is defined
so that it is invariant under the action of $G$ on $\pflags$: for any
$x,y$ in some proper domain $\Omega \subset \pflags$, and any
$g \in G$, we have
\begin{equation}
  \label{eq:g_invariance}
  C_\Omega(x,y) = C_{g\Omega}(gx, gy).
\end{equation}

In general, $C_\Omega$ is not a complete metric. However, $C_\Omega$
induces the standard topology on $\Omega$ as an open subset of
$\flags$. We will show that for a $\mc{G}$-path $\{\alpha_n\}$, the
diameter of
\[
  \alpha_1 \cdots \alpha_n U_{v_{n+1}}
\]
with respect to $C_{U_{v_1}}$ tends to zero as $n \to \infty$.

Zimmer's construction of $C_\Omega$ depends on an irreducible
representation $\zeta:G \to \PGL(V)$ for some real vector space
$V$. This is provided by a theorem of
Gu\'eritaud-Guichard-Kassel-Wienhard.
\begin{theorem}[\cite{ggkw2017anosov}, see also
  \cite{zimmer2018proper}, Theorem 4.6]
  \label{thm:flag_irrep_embedding}
  There exists a real vector space $V$, an irreducible representation
  $\zeta:G \to \PGL(V)$, a line $\ell \subset V$, and a hyperplane
  $H \subset V$ such that:
  \begin{enumerate}
  \item $\ell + H = V$.
  \item The stabilizer of $\ell$ in $G$ is $\pargroup$ and the
    stabilizer of $H$ in $G$ is $\oppgroup$.
  \item $g\pargroup g^{-1}$ and $h\oppgroup h^{-1}$ are opposite if
    and only if $\zeta(g)\ell$ and $\zeta(h)H$ are transverse.
  \end{enumerate}
\end{theorem}

The representation $\zeta$ determines a pair of embeddings
$\iota:\pflags \to \P(V)$ and $\iota^*:\oppflags \to \P(V^*)$ by
\[
  \iota(g\pargroup) = \zeta(g)\ell, \qquad \iota^*(g\oppgroup) =
  \zeta(g)H.
\]
In this section, we will identify $\P(V^*)$ with the space of
projective hyperplanes in $\P(V)$, by identifying the projectivization
of a functional $w \in V^*$ with the projectivization of its kernel.

\begin{definition}
  Let $\Omega$ be an open subset of $\pflags$. The \emph{dual domain}
  $\Omega^* \subset \oppflags$ is
\[
  \Omega^* = \{\nu \in \oppflags : \nu \textrm{ is opposite to }\xi
  \textrm{ for every } \xi \in \overline{\Omega}\}.
\]
Note that $\Omega^*$ is open if and only if $\Omega$ is a proper
domain.
\end{definition}

\begin{definition}
  Let $w_1, w_2 \in \P(V^*)$, and let $z_1, z_2 \in \P(V)$. The
  \emph{cross-ratio} $[w_1, w_2 ; z_1, z_2]$ is defined by
  \[
    \frac{\tilde{w}_1(\tilde{z}_2)\tilde{w}_2(\tilde{z}_1)}
    {\tilde{w}_1(\tilde{z}_1)\tilde{w}_2(\tilde{z}_2)},
  \]
  where $\tilde{w}_i$, $\tilde{z}_i$ are respectively lifts of $w_i$
  and $z_i$ in $V^*$ and $V$.
\end{definition}

\begin{remark}
  When $V$ is two-dimensional, we can identify the projective line
  $\P(V^*)$ with $\P(V)$ by identifying each $[w] \in \P(V^*)$ with
  $[\ker(w)] \in \P(V)$. In that case, the cross-ratio defined above
  agrees with the standard four-point cross-ratio on $\RP^1$, given by
  \begin{equation}
    \label{eq:cross_ratio_definition}
    [a, b; c, d] := \frac{(d - a)(c - b)}{(c - a)(d - b)}.
  \end{equation}
  The differences in (\ref{eq:cross_ratio_definition}) can be measured
  in any affine chart in $\RP^1$ containing $a, b, c, d$. Our
  convention is chosen so that if we identify $\RP^1$ with
  $\R \cup \{\infty\}$, we have $[0, \infty; 1, z] = z$.
\end{remark}

\begin{definition}
  Let $\Omega \subset \pflags$ be a proper domain. We define the
  function $C_\Omega:\Omega \times \Omega \to \R$ by
  \[
    C_\Omega(x,y) = \sup_{\xi_1, \xi_2 \in
      \Omega^*}\log|[\iota^*(\xi_1), \iota^*(\xi_2); \iota(x),
    \iota(y)]|.
  \]
\end{definition}
For any $g \in G$ and any proper domain $\Omega \subset \pflags$, we
have $(g\Omega)^* = (g\Omega^*)$. So $C_\Omega$ must satisfy the
$G$-invariance condition (\ref{eq:g_invariance}).

If $\Omega$ is a properly convex subset of $\P(V)$, and $\zeta$,
$\iota$, $\iota^*$ are the identity maps on $\PGL(V)$, $\P(V)$, and
$\P(V^*)$ respectively, then $C_\Omega$ agrees with the \emph{Hilbert
  metric} on $\Omega$ (see \Cref{defn:hilbert_metric}). More generally
we have:

\begin{theorem}[\cite{zimmer2018proper}, Theorem 5.2]
  If $\Omega$ is open and bounded in an affine chart, then $C_\Omega$
  is a metric on $\Omega$ which induces the standard topology on
  $\Omega$ as an open subset of $\pflags$.
\end{theorem}

\begin{remark}
  This particular result in \cite{zimmer2018proper} is stated only for
  noncompact simple Lie groups, but the proof only assumes that $G$ is
  semisimple with no compact factor.
\end{remark}

Since taking duals of proper domains reverses inclusions, it follows
that if $\Omega_1 \subset \Omega_2$, then
$C_{\Omega_1} \ge C_{\Omega_2}$.  Our goal now is to sharpen this
inequality, and show:
\begin{prop}
  \label{prop:caratheodory_metric_shrinks}
  Let $\Omega_1$, $\Omega_2$ be proper domains in $\pflags$, such that
  $\overline{\Omega_1} \subset \Omega_2$.

  There exists a constant $\lambda > 1$ (depending on $\Omega_1$ and
  $\Omega_2$) so that for all $x,y \in \Omega_1$,
  \[
    C_{\Omega_1}(x, y) \ge \lambda \cdot C_{\Omega_2}(x,y).
  \]
\end{prop}

A consequence is the following, which in particular implies
\Cref{prop:compatible_system_in_flags_contracting}.
\begin{cor}
  \label{cor:exponential_path_shrinking}
  Let $\mc{G}$ be a $\Gamma$-graph for a relatively hyperbolic group
  $\Gamma$, and let $\{U_v\}$ be a $\mc{G}$-compatible system of open
  subsets of $\pflags$. If each $U_v$ is a proper domain, then there
  are constants $\lambda_1, \lambda_2 > 0$ so that for any
  $\mc{G}$-path $\{\alpha_n\}$ in the $\Gamma$-graph $\mc{G}$, the
  diameter of
  \[
    \alpha_1 \cdots \alpha_n \cdot U_{v_{n+1}}
  \]
  with respect to $C_{U_{v_1}}$ is at most
  \[
    \lambda_1 \cdot \exp(-\lambda_2 \cdot n).
  \]
\end{cor}
\begin{proof}
  For any open set $U \subset \pflags$ and $A \subset U$, we let
  $\mr{diam}_U(A)$ denote the diameter of $A$ with respect to the
  metric $C_U$. We choose a uniform $\eps > 0$ so that in some fixed
  metric on $\pflags$, every edge $(v, w)$ in $\mc{G}$, and every
  $\alpha \in T_v$, we have
  \[
    \alpha N(U_w, \eps) \subset U_v.
  \]
  Then for each vertex set $U_v$, we write $U_v^\eps = N(U_v, \eps)$.

  We take
  \[
    \lambda_1 = \max\{\mr{diam}_{U_v^\eps}(U_v)\}.
  \]
  \Cref{prop:caratheodory_metric_shrinks} implies that there exists
  $\lambda_v > 0$ such that for all $x, y \in U_v$, we have
  \[
    C_{U_v}(x,y) \ge \exp(\lambda_v) \cdot C_{U_v^\eps}(x, y).
  \]
  Take $\lambda_2 = \min_{v} \{\lambda_v\}$. We claim that for all
  $n \ge 1$, we have
  \[
    \mr{diam}_{U_1^\eps}(\alpha_1 \cdots \alpha_nU_{v_n}) \le \lambda_1
    \exp(-\lambda_2 \cdot (n - 1)).
  \]
  We prove the claim via induction on the length of the $\mc{G}$-path
  $\{\alpha_n\}$. For $n = 1$, the claim is true because
  $\alpha_1U_{v_2} \subset U_{v_1}$. For $n > 1$, we can assume
  \[
    \lambda_1 \exp(-\lambda_2(n - 2)) \ge
    \mr{diam}_{U_{v_2}^\eps}(\alpha_2 \cdots \alpha_n \cdot
    U_{v_{n+1}}).
  \]
  Then we have
  \begin{align*}
    \mr{diam}_{U_{v_2}^\eps}(\alpha_2 \cdots \alpha_n \cdot
    U_{v_{n+1}}) &= \mr{diam}_{\alpha_1U_{v_2}^\eps}(\alpha_1 \cdots \alpha_n \cdot
                   U_{v_{n+1}})\\
                 &\ge \mr{diam}_{U_{v_1}}(\alpha_1 \cdots \alpha_n \cdot
                   U_{v_{n+1}})\\
                 &\ge \exp(\lambda_2) \cdot
                   \mr{diam}_{U_{v_1}^\eps}(\alpha_1 \cdots \alpha_n
                   \cdot U_{v_{n+1}}).
  \end{align*}
  Finally, the claim implies the corollary because we know that
  \begin{align*}
    \mr{diam}_{U_1}(\alpha_1 \cdots \alpha_nU_{n+1})
    &\le \mr{diam}_{\alpha_1 U_2^\eps}(\alpha_1 \cdots \alpha_nU_{n+1}) \\
    &= \mr{diam}_{U_2^\eps}(\alpha_2 \cdots \alpha_nU_{n+1})\\
    &\le \lambda_1 \exp(\lambda_2(n - 2)).
  \end{align*}
  So, we can replace $\lambda_1$ with $\lambda_1\exp(-2\lambda_2)$ to
  get the desired result.
\end{proof}

We now proceed with the proof of
\Cref{prop:caratheodory_metric_shrinks}. We first observe that in the
special case where $\Omega_1, \Omega_2$ are properly convex subsets of
real projective space, one can show the desired result essentially via
the following:
\begin{prop}
  \label{prop:hilbert_metric_contracts}
  Let $a, b, c, d$ be points in $\RP^1$, arranged so that
  $a < b < c < d \le a$ in a cyclic ordering on $\RP^1$. Then there
  exists a constant $\lambda > 1$, depending only on the cross-ratio
  $[a, b; c, d]$, so that for all distinct $x, y \in (b, c)$, we have
  \[
    |\log[b,c; x,y]| \ge \lambda \cdot |\log[a,d; x, y]|.
  \]
\end{prop}
\Cref{prop:hilbert_metric_contracts} is a standard fact in real
projective geometry and can be verified by a computation. Note that we
allow the degenerate case $a = d$: in this situation the right-hand
side is identically zero for distinct $x, y \in (b,c)$. We allow no
other equalities among $a,b,c,d$, so the cross-ratio $[a,b; c,d]$ lies
in $\R - \{1\}$.

To apply \Cref{prop:hilbert_metric_contracts} to our situation, we
need to get some control on the behavior of the embeddings
$\iota:\pflags \to \P(V)$ and $\iota^*:\oppflags \to \P(V^*)$. We do
so in the next three lemmas below.

\begin{lem}
  \label{lem:subgroup_acts_secant_line}
  Let $x,y$ be distinct points in $\pflags$. There exists a
  one-parameter subgroup $g_t \subset G$ such that $\zeta(g_t)$ fixes
  $\iota(x)$ and $\iota(y)$, and acts nontrivially on the projective
  line $L_{xy}$ spanned by $\iota(x)$ and $\iota(y)$.
\end{lem}
\begin{proof}
  We can write $x = gP^+$ for some $g \in G$. Let $\mf{a}$ denote a
  Cartan subalgebra of the Lie algebra $\mf{g}$ of $G$. There is a
  conjugate $\mf{a}'$ of $\mf{a}$ such that the action of
  $\exp(\mf{a}')$ on $\pflags$ fixes both $x$ and $y$ (see
  \cite{eberlein}, Proposition 2.21.14). So, up to the action of $G$
  on $\pflags$, we can assume that $x$ is fixed by a standard
  parabolic subgroup $\stdpargroup$ conjugate to $P^+$, and that
  $x, y$ are both fixed by the Cartan subgroup $\exp(\mf{a})$.

  We choose $Z \in \mf{a}^+$ so that $\alpha(Z) \ne 0$ for all
  $\alpha \in \theta$. Then $g_t = \exp(tZ)$ is a 1-parameter subgroup
  of $G$ fixing $x$. As $t \to +\infty$, $g_t$ is
  $\stdpargroup$-divergent, and converges to the constant map $x$
  uniformly on compacts in $\Opp(x)$.

  Then \cite{ggkw2017anosov}, Lemma 3.7 implies that $\zeta(g_t)$ is
  $P_1$-divergent, where $P_1$ is the stabilizer of a line in $V$, and
  $\iota(x)$ is the unique one-dimensional eigenspace of $\zeta(g_t)$
  whose eigenvalue has largest modulus. And, since $\zeta(g_t)$ fixes
  $\iota(x)$ and $\iota(y)$, $\zeta(g_t)$ preserves $L_{xy}$, and acts
  nontrivially since the eigenvalues of $\zeta(g_t)$ on $\iota(x)$ and
  $\iota(y)$ must be distinct.
\end{proof}

\begin{lem}
  \label{lem:subgroup_acts_tangent_line}
  Let $L$ be any projective line in $\P(V)$ tangent to the image of
  the embedding $\iota:\pflags \to \P(V)$ at a point $\iota(x)$ for
  $x \in \pflags$. There exists a one-parameter subgroup $g_t$ of $G$
  so that $\zeta(g_t)$ acts nontrivially on $L$ with unique fixed
  point $\iota(x)$.
\end{lem}
\begin{proof}
  Fix a sequence $y_n \in \pflags$ such that $y_n \ne x$ and the
  projective line $L_n$ spanned by $\iota(x)$ and $\iota(y_n)$
  converges to $L$. By \Cref{lem:subgroup_acts_secant_line}, there
  exists $Z_n \in \mf{g}$ so that $\zeta(\exp(tZ_n))$ acts
  nontrivially on $L_n$, with fixed points $\iota(x)$ and
  $\iota(y_n)$.

  In the projectivization $\P(\mf{g})$, $[Z_n]$ converges to some
  $[Z]$. Since $\zeta:G \to \PGL(V)$ has finite kernel, there is an
  induced map $\zeta:\P(\mf{g}) \to \P(\mf{sl}(V))$, which satisfies
  \[
    \zeta([Z_n]) \to \zeta([Z]).
  \]
  A continuity argument shows that the one-parameter subgroup
  $\zeta(\exp(tZ))$ acts nontrivially on the line $L$, and has unique
  fixed point at $\iota(x)$.
\end{proof}

\begin{lem}
  \label{lem:line_intersections_open}
  Let $\Omega \subset \pflags$ be a proper domain, and let $L$ be a
  projective line in $\P(V)$ which is either spanned by two points in
  $\iota(\Omega)$, or is tangent to $\iota(\pflags)$ at a point
  $\iota(x)$ for $x \in \Omega$. Then
  \[
    W_L = \{v \in L : v = \iota^*(\xi) \cap L \textrm{ for } \xi \in
    \Omega^*\}
  \]
  is a nonempty open subset of $L$.
\end{lem}
\begin{proof}
  $W_L$ is nonempty since $\Omega^*$ is nonempty. So let $v \in W_L$,
  and choose $\xi \in \Omega^*$ so that $\iota^*(\xi) \cap L = v$. We
  need to show that an open interval $I \subset L$ containing $v$ is
  contained in $W_L$.

  If $L$ is spanned by $x, y \in \iota(\Omega)$, then
  \Cref{lem:subgroup_acts_secant_line} implies that we can find a
  one-parameter subgroup $g_t \in G$ such that $\zeta(g_t)$ fixes $x$
  and $y$, and acts nontrivially on $L$. Since $\Omega^*$ is open, we
  can find $\eps > 0$ so that $g_t \cdot \xi \in \Omega^*$ for
  $t \in (-\eps, \eps)$. Since $x$ and $y$ are in $\iota(\Omega)$,
  $\iota^*(\xi)$ is transverse to both $x$ and $y$, so we have
  $v \ne x$, $v \ne y$. Then as $t$ varies from $-\eps$ to $\eps$,
  \[
    \iota^*(g_t \cdot \xi) \cap L = \zeta(g_t) \cdot v
  \]
  gives an open interval in $W_L$ containing $v$.

  A similar argument using \Cref{lem:subgroup_acts_tangent_line} shows
  that the claim also holds if $L$ is tangent to $\iota(\Omega)$.
\end{proof}

We can now prove a slightly weaker version of
\Cref{prop:caratheodory_metric_shrinks}, which we will then use to
show the stronger version.
\begin{lem}
  \label{lem:cpct_caratheodory_bound}
  Let $\Omega_1, \Omega_2$ be proper domains in $\pflags$, with
  $\overline{\Omega_1} \subset \Omega_2$, and let $K \subset \Omega_1$
  be compact. There exists a constant $\lambda > 1$ such that for all
  $x, y \in K$,
  \[
    C_{\Omega_1}(x,y) \ge \lambda \cdot C_{\Omega_2}(x,y).
  \]
\end{lem}
\begin{proof}
  Since $K$ is compact, it suffices to show that for fixed $x \in
  \Omega_1$, the ratio
  \[
    \frac{C_{\Omega_1}(x,y)}{C_{\Omega_2}(x,y)}
  \]
  is bounded below by some $\lambda > 1$ as $y$ varies in $K - \{x\}$.

  Let $y \in K - \{x\}$, and let $L_{xy}$ denote the projective line
  spanned by $\iota(x)$ and $\iota(y)$. Choose
  $\xi, \eta \in \overline{\Omega_2^*}$ so that
  \[
    C_{\Omega_2}(x,y) = \log|[\iota^*(\xi), \iota^*(\eta); \iota(x),
    \iota(y)]|.
  \]
  That is, if $v = \iota^*(\xi) \cap L_{xy}$,
  $w = \iota^*(\eta) \cap L_{xy}$, we have
  \[
    C_{\Omega_2}(x,y) = \log|[v, w; \iota(x), \iota(y)]| =
    \log\frac{|v - \iota(y)|\cdot|w - \iota(x)|}{|v - \iota(x)|\cdot|w
      - \iota(y)|},
  \]
  where the distances are measured in any identification of $L_{xy}$
  with $\RP^1 = \R \cup \{\infty\}$.

  We can choose an identification of $L_{xy}$ with
  $\R \cup \{\infty\}$ so that either $v < \iota(x) < \iota(y) < w$ or
  $v < \iota(x) < w < \iota(y)$. In either case, for any
  $v' \in (v, \iota(x)) \subset L_{xy}$, we have
  \[
    \log|[v', w; \iota(x), \iota(y)]| > \log|[v, w; \iota(x),
    \iota(y)]|.
  \]

  We know that $\overline{\Omega_2^*} \subset \Omega_1^*$, so
  $\xi, \eta$ lie in $\Omega_1^*$. Then
  \Cref{lem:line_intersections_open} implies that there exists
  $\xi' \in \Omega_1^*$ so that $v' = \iota^*(\xi') \cap L_{xy}$ lies
  in the interval $(v, \iota(x)) \subset L_{xy}$. See
  \Cref{fig:nested_flag_nbhds}.

  \begin{figure}[h]
    \centering
   
    \def14cm{14cm}
    \import{svg-inkscape/}{nested_flag_nbhds_disconnected_svg-tex.pdf_tex}
    \caption{We can always find $\xi' \in \Omega_1^*$ close to $\xi$
      so that the absolute value of the cross-ratio
      $[\iota^*(\xi), \iota^*(\nu); \iota(x), \iota(y)]$ increases
      when we replace $\xi$ with $\xi'$. In particular this is
      possible even when the sets $\iota(\Omega_1)$, $\iota(\Omega_2)$
      fail to be convex (left) or even connected
      (right).}\label{fig:nested_flag_nbhds}
  \end{figure}
  
  Then, we have
  \begin{align*}
    C_{\Omega_1}(x,y) &\ge \log|[\iota^*(\xi'), \iota^*(\eta) ;
                        \iota(x), \iota(y)]|\\
                      &= \log|[v', w; \iota(x), \iota(y)]\\
                      &> \log|[v, w; \iota(x), \iota(y)]\\
                      &= C_{\Omega_2}(x,y).
  \end{align*}
  This shows that $\frac{C_{\Omega_1}(x,y)}{C_{\Omega_2}(x,y)} > 1$
  for all $y \in K - \{x\}$. We still need to find some uniform
  $\lambda > 1$ so that
  $\frac{C_{\Omega_1}(x,y)}{C_{\Omega_2}(x,y)} \ge \lambda$ for all
  $y \in K - \{x\}$. To see this, suppose for the sake of a
  contradiction that for a sequence $y_n \in K - \{x\}$, we have
  \begin{equation}
    \label{eq:caratheodory_ratios_bounded}
    \frac{C_{\Omega_1}(x,y_n)}{C_{\Omega_2}(x, y_n)} \to 1.
  \end{equation}
  Since $K$ is compact, $y_n$ must converge to $x$. Up to subsequence,
  the sequence of projective lines $L_n$ spanned by $\iota(x)$ and
  $\iota(y_n)$ converges to a line $L$ tangent to $\iota(\pflags)$ at
  $\iota(x)$.

  For each $y_n$, choose $\xi_n$, $\eta_n \in \overline{\Omega_2^*}$
  so that
  \[
    C_{\Omega_2}(x, y_n) = \log|[\iota^*(\xi_n), \iota^*(\eta_n) ;
    \iota(x), \iota(y_n)]|.
  \]
  Let $v_n = \iota^*(\xi_n) \cap L_n$,
  $w_n = \iota^*(\eta_n) \cap L_n$. Then up to subsequence $\xi_n$
  converges to $\xi \in \overline{\Omega_2^*}$, $\eta_n$ converges to
  $\eta \in \overline{\Omega_2^*}$, and $v_n$, $w_n$ respectively
  converge to $v = \iota^*(\xi) \cap L$, $w = \iota^*(\eta) \cap L$.

  Since $x$ is in $\Omega_2$, $\iota^*(\xi)$ and $\iota^*(\eta)$ are
  both transverse to $\iota(x)$---so in particular $x \ne w$ and
  $x \ne v$ (although a priori we could have $v = w$).

  Since $\xi \in \overline{\Omega_2^*} \subset \Omega_1^*$,
  \Cref{lem:line_intersections_open} implies that there exist
  $\xi', \eta' \in \Omega_1^*$ so that for some identification of $L$
  with $\R \cup \{\infty\}$, we have
  \[
    v < \iota^*(\xi') \cap L < \iota(x) < \iota^*(\eta') \cap L < w.
  \]
  Note that this is possible even if $v = w$, because then we can just
  identify both $v$ and $w$ with $\infty$. Let
  $v_n' = \iota^*(\xi') \cap L_n$, and let
  $w_n' = \iota^*(\eta') \cap L_n$. Respectively, $v_n'$ and $w_n'$
  converge to $v' = \iota^*(\xi') \cap L$ and
  $w' = \iota^*(\eta') \cap L$.

  This means that the cross-ratios $[v_n, v_n'; w_n', w_n]$ converge
  to $[v, v'; w', w] \in \R - \{1\}$, and in particular are bounded
  away from both $\infty$ and $1$ for all $n$.

  We choose identifications of $L_n$ with $\R \cup \{\infty\}$ so that
  $v_n < v_n' < \iota(x) < w_n' < w_n$. Since $y_n$ converges to $x$,
  for all sufficiently large $n$, we have $v_n' < \iota(y_n) <
  w_n'$. Then, \Cref{prop:hilbert_metric_contracts} implies that for
  all $n$, we have
  \[
    \log |[v_n', \iota(x), \iota(y_n), w_n']| \ge \lambda \cdot
    \log|[v_n, \iota(x), \iota(y_n), w_n]|
  \]
  for some $\lambda > 1$ independent of $n$. But then since
  \[
    C_{\Omega_1}(x, y_n) \ge \log|[\iota^*(\xi'), \iota^*(\eta');
    \iota(x), \iota(y_n)]|,
  \]
  we have $C_{\Omega_1}(x,y_n) / C_{\Omega_2}(x,y_n) \ge \lambda$ for
  all $n$, contradicting (\ref{eq:caratheodory_ratios_bounded}) above.
\end{proof}

\begin{proof}[Proof of \Cref{prop:caratheodory_metric_shrinks}]
  We fix an open set $\Omega_{1.5}$ such that
  $\overline{\Omega_1} \subset \Omega_{1.5}$ and
  $\overline{\Omega_{1.5}} \subset \Omega_2$. Since
  $C_{\Omega_1}(x,y) \ge C_{\Omega_{1.5}}(x,y)$ for all
  $x, y \in \Omega_1$, we just need to see that there is some
  $\lambda > 1$ so that
  \[
    \frac{C_{\Omega_{1.5}}(x,y)}{C_{\Omega_2}(x,y)} \ge \lambda
  \]
  for all $x, y \in \overline{\Omega_1}$. This follows from
  \Cref{lem:cpct_caratheodory_bound}.
\end{proof}

\subsection{Contracting paths are $\pargroup$-divergent}

\begin{prop}
  \label{lem:graph_contracting_implies_contracting}
  Let $\mc{G}$ be a $\Gamma$-graph for a group $\Gamma \subset G$, and
  let $\{U_v\}$ be a $\mc{G}$-compatible system of open sets of
  $\pflags$ with each $U_v$ a proper domain.

  If $\alpha_n$ is a contracting $\mc{G}$-path, then the sequence
  \[
    \gamma_n = \alpha_1 \cdots \alpha_n
  \]
  is $\pargroup$-divergent with unique limit point $\xi$, where
  $\{\xi\} = \bigcap_{n=1}^\infty \gamma_n U_{n+1}$.
\end{prop}
\begin{proof}
  Consider the sequence of open sets
  \[
    \gamma_n \cdot U_{v_{n+1}}.
  \]
  Up to subsequence, $U_{v_{n+1}}$ is a fixed open set
  $U \subset \pflags$. By assumption $\{\alpha_n\}$ is a contracting
  path, so $\gamma_n \cdot U_{v_{n+1}}$ converges to a singleton
  $\{\xi\}$. So, we apply
  \Cref{prop:local_contracting_implies_global_contracting}.
\end{proof}


\subsection{Generalized horospheres}

In \cite{clt2018deforming}, Cooper-Long-Tillmann show that if $C$ is a
generalized cusp with holonomy $\rho:\pi_1C \to \PGL(d+1,\R)$, there
is a finite-index subgroup $\Gamma_1 \subseteq \pi_1C$ (depending only
on $\pi_1C$ and $d$) so that $\rho(\Gamma_1)$ is a lattice in a
\emph{syndetic hull} of $\rho(\Gamma_1)$: a uniquely determined
connected Lie group $T(\rho) \subset \PGL(d+1, \R)$, conjugate into
the group of upper triangular matrices. This group is called the
\emph{translation group} of the cusp.

We may assume that $\Gamma_1$ is free abelian, so it is a lattice in
$\Gamma_1 \ten_\Z \R \simeq \R^{d-1}$. The restriction
\[
  \rho|_{\Gamma_1}:\Gamma_1 \to \PGL(d+1, \R)
\]
extends uniquely to an embedding of Lie groups
\[
  \iota_\rho:\R^{d-1} \hookrightarrow \PGL(d+1, \R)
\]
with image $T(\rho)$.

We observe the following:
\begin{prop}
  \label{prop:cusp_embedding_c1_continuous}
  Let $C$ be a generalized cusp. The embedding
  \[
    \iota_\rho:\R^{d-1} \to \PGL(d+1, \R)
  \]
  varies continuously with $\rho \in \vfg(C)$ in the compact-open
  topology on maps from $\R^{d-1}$ into $\PGL(d+1, \R)$.
\end{prop}
\begin{proof}
  The Lie algebra of $\PGL(d+1, \R)$ is identified with
  $\mf{sl}(d+1, \R)$. Fix a finite set $S$ of generators for
  $\Gamma_1$. The subspace $\mf{s}_\rho \subset \mf{sl}(d+1, \R)$
  spanned by $\log(\rho(S))$ varies smoothly with $\rho$, and the
  induced map
  \[
    \R^{d-1} \to \mf{sl}(d+1, \R)
  \]
  with image $\mf{s}_\rho$ varies continuously in the compact-open
  topology. The embedding $\iota_\rho$ is given by composition with
  the exponential map.
\end{proof}

When $C = \Omega / \Gamma$ is projectively equivalent to a cusp in
some hyperbolic manifold, we can assume that $\Omega$ is a closed
horoball in $\H^d$. The nonideal boundary $\dee_n\Omega$ is a
horosphere preserved by $\Gamma$, which carries a $\Gamma$-invariant
Euclidean metric. In this case the translation group is the group of
Euclidean translations on $\dee_n\Omega$.

When $C$ is a generalized cusp, we can always find some orbit of the
translation group $T(\rho)$ in $\RP^d$ which is a strictly convex
hypersurface (see Proposition 6.22 in \cite{clt2018deforming}). This
hypersurface is called a \emph{generalized horosphere}. Its convex
hull in $\RP^d$ is a \emph{generalized horoball}. We can always choose
this horoball so that its quotient by $\Gamma$ is contained in the
generalized cusp $C$.

Now consider a convex projective manifold $M = \Omega / \Gamma$ with
strictly convex boundary which can be written as a union of a compact
piece and finitely many generalized cusps. It is always possible to
choose the cusps so that each cusp $C \subset M$ is a quotient
$C = \Omega_C / \Gamma_C$, where $\Omega_C \subset \Omega$ is a convex
subset whose nonideal boundary is a generalized horosphere, and
$\Gamma_C = \pi_1C$ is the cusp group. The boundary $\dee_n\Omega_C$
is \emph{homogeneous}: the translation group $T(\Gamma_C)$ acts simply
transitively on $\dee_n\Omega_C$.

\subsection{The ideal boundary}

The Ballas-Cooper-Leitner classification of generalized cusps allows
us to get a more explicit description of the ideal boundary of
$\Omega$. Given a generalized cusp $C = \Omega / \Gamma$, we let
$\Omega_C$ denote the ``standard'' $\Gamma$-invariant domain with
homogeneous nonideal boundary, alluded to above.

\begin{prop}[Lemmas 1.24 and 1.25 in \cite{bcl2020generalized}]
  Let $C = \Omega_C / \Gamma$ be a generalized cusp. The ideal
  boundary of $\Omega_C$ is a projective $k$-simplex $\Delta_C$. There
  is a unique supporting hyperplane $H_C$ of $\Omega_C$ containing
  $\Delta_C$, and the affine chart
  \[
    \A_C = \RP^d - H_C
  \]
  is the unique affine chart containing $\Omega_C$ as a closed subset.
\end{prop}

The vertices of $\Delta_C$ must be preserved by $\Gamma$, and in fact
they are all eigenvectors for the translation group $T(\Gamma)$.

Each generalized horosphere $S_C$ for $C$ is a strictly convex
hypersurface contained in the affine chart $\A_C$. The closure of this
hypersurface in $\RP^d$ is either $S_C \cup \dee\Delta_C$ (if $C$ is a
``type $d$'' cusp) or $S_C \cup \Delta_C$ (if $C$ is any other type of
cusp).

\subsection{Deformations of convex hypersurfaces}

The main ingredient in the proof of
\Cref{thm:finite_volume_projective_deformations_stable} is the
following:

\begin{lem}
  \label{lem:convex_hypersurface_deformation}
  Let $C$ be a hyperbolic cusp with holonomy $\rho$. Let $p_C$ be the
  cusp point, and let $H_C$ be the unique supporting hyperplane of
  $\Omega_C$ at $p_C$.

  Let $x \in \A_C = \RP^d - H_C$, let $U \subset \RP^d$ be an open
  subset containing $p_C$, and let $F \subset \pi_1C$ be a cofinite
  subset such that $\rho(F) \cdot x \subset U$.

  There exists a neighborhood $\mc{W}$ of $\rho$ in $\vfg(C)$ so that
  for any $\rho' \in \mc{W}$, we have
  \[
    \rho'(F) \cdot x \subset U.
  \]
\end{lem}
\begin{proof}
  \cite{clt2018deforming}, Theorem 6.25 implies that we can choose a
  neighborhood $\mc{W}$ of $\rho$ in $\vfg(C)$ consisting of
  holonomies of generalized cusps. For any $\rho' \in \mc{W}$, we let
  $\Omega'$ denote a ``standard'' properly convex set preserved by
  $\rho'(\pi_1C)$, whose non-ideal boundary is a generalized
  horosphere.

  Since $p_C$ and $H_C$ are respectively the unique eigenvector and
  fixed hyperplane of $\rho(\pi_1C)$, we can choose our neighborhood
  $\mc{W}$ so that for any $\rho' \in \mc{W}$, any eigenvectors and
  fixed hyperplanes of $\rho'(\pi_1C)$ are close to $p_C$, $H_C$.

  In particular, we can choose $\mc{W}$ so that the ideal boundary of
  $\Omega'$ is a $k$-simplex $\Delta'$ contained in $U$. And, by
  applying a small conjugation in $\PGL(d+1, \R)$, we can assume that
  $H_C$ is the unique supporting hyperplane of $\Omega'$ containing
  $\Delta'$.

  Let $T(\rho)$ be the translation group of $\rho$. The orbit
  $T(\rho) \cdot x$ is a paraboloid in $\A_C$. We can write
  $\A_C = \R^{d-1} \times \R$, and then view this paraboloid as the
  graph of a function
  \[
    f_\rho:\R^{d-1} \to \R.
  \]
  The function $f_\rho$ is determined by the condition
  \[
    (u, f_\rho(u)) = \iota_\rho(u) \cdot x.
  \]
  Here $\iota_\rho:\R^{d-1} \to \PGL(d+1, \R)$ restricts to $\rho$ on
  a finite-index subgroup $\Gamma_1 \subset \pi_1C$, with $\Gamma_1$
  identified with $\Z^{d-1} \subset \R^{d-1}$.

  If $T(\rho')$ is the translation group of $\rho'$, then Lemma 6.24
  in \cite{clt2018deforming} implies that $T(\rho') \cdot x$ is a
  strictly convex hypersurface $S' \subset \A_C$. The hypersurface
  $S'$ is the graph of a map $f_{\rho'}:\R^{d-1} \to \R$, satisfying
  \[
    (u, f_{\rho'}(u)) = \iota_{\rho'}(u) \cdot x.
  \]
  \Cref{prop:cusp_embedding_c1_continuous} implies that $f_{\rho'}$
  varies continuously (in the compact-open topology on continuous maps
  $\R^{d-1} \to \R$) as $\rho'$ varies in $\mc{W}$.

  We fix a norm $||\cdot||$ on $\R^{d-1}$. There is a constant $D$ so
  that for any $(u, v) \in \A_C$, if $||u|| > 1$ and
  $|v| / ||u|| > D$, then $(u, v)$ is contained in the neighborhood
  $U$ of $p_C$.

  $f_{\rho'}$ is a strictly convex function, which we can assume is
  nonnegative and uniquely minimized at the origin. So, if
  $f_{\rho'}(u) / ||u|| > D$ on $\{u \in \R^{d-1} : ||u|| = M\}$ for
  some constant $M$, then $f_{\rho'}(u) / ||u|| > D$ for all $u$ with
  $||u|| \ge M$.

  So, as long as $W$ is sufficiently small, there is a fixed ball
  $B \subset \R^{d-1}$ so that if $u \in \R^{d-1} - B$, then
  \[
    (u, f_{\rho'}(u)) \in U
  \]
  for any $\rho' \in \mc{W}$.

  The ball $B$ contains at most finitely many elements of
  $\Gamma_1 - F$. So we can in fact choose $\mc{W}$ small enough so
  that for any $\rho' \in \mc{W}$, every element of
  $\rho'(\Gamma_1 - F) \cdot x$ lies in $U$. Then since
  $|\Gamma_1 : \pi_1C| < \infty$ we can shrink $\mc{W}$ even further
  to guarantee that for any $\rho' \in \mc{W}$,
  \[
    \rho'(\pi_1C - F) \cdot x
  \]
  lies in $U$ as well.
\end{proof}

\subsection{Peripheral stability}

\begin{proof}[Proof of
  \Cref{thm:finite_volume_projective_deformations_stable}]
  Let $\rho:\pi_1M \to \PGL(d+1, \R)$ be the holonomy of a
  finite-volume convex projective manifold $M$, and let $\Omega$ be a
  $\rho$-invariant strictly convex domain such that
  $M = \Omega / \rho(\pi_1M)$. Write $\Gamma = \pi_1M$, and let
  $\mc{H}$ be the collection of cusp groups, so $(\Gamma, \mc{H})$ is
  a relatively hyperbolic pair, and for each $H \in \mc{H}$, $\rho|_H$
  is the holonomy of a hyperbolic cusp.

  Since $\Omega$ is strictly convex, \cite{clt2015convex}, Theorem
  11.6 also implies that $\Omega$ has $C^1$ boundary. So there is a
  $\rho$-equivariant homeomorphism $\dee \Omega \to \dee \Omega^*$
  assigning the point $z \in \dee \Omega$ to the unique supporting
  hyperplane of $\Omega$ at $z$. We let $\dee \hat{\Omega}$ denote the
  space
  \[
    \{(x, w) \in \projflags : x \in \dee \Omega, w \in \dee
    \Omega^*\}.
  \]
  
  There is an equivariant homeomorphism
  $\psi:\bgamh \to \dee \hat{\Omega}$, with the
  parabolic points in $\bgamh$ corresponding to the
  fixed flags of the cusp groups. The inverse map
  $\phi:\dee\hat{\Omega} \to \bgamh$ extends the
  convergence dynamics of $\Gamma$. For each parabolic point
  $p \in \bgamh$, the open set $C_p$ is
  \[
    \Opp(\psi(p)) = \{\xi \in \projflags : \xi \textrm{ is opposite to
    } \psi(p)\}.
  \]

  Let $\pi:\projflags \to \RP^d$ be the canonical projection map. It
  suffices to show that for any compact set $K \subset \pi(C_p)$, any
  open neighborhood $U$ of $\pi(\psi(p))$ in $\RP^d$, and any cofinite
  subset $F \subset \Gamma_p = \Stab_\Gamma(p)$ such that
  \[
    \rho(F) \cdot K \subset U,
  \]
  we can find a neighborhood $\mc{W} \subset \vfg(\Gamma_p)$
  containing $\rho$ such that
  \[
    \rho'(F) \cdot K \subset U
  \]
  for any $\rho' \in \mc{W}$. (The same argument applied dually will
  show that we can upgrade $K$ to a compact subset of
  $C_p \subset \projflags$ and $U$ to an open subset in $\projflags$).

  For any $x$ in such a compact set $K \subset \pi(C_p)$,
  \Cref{lem:convex_hypersurface_deformation} implies that there is an
  open neighborhood $\mc{W} \subset \vfg(\Gamma_p)$, containing the
  restriction $\rho|_{\Gamma_p}$, so that for any $\rho' \in \mc{W}$,
  we have
  \[
    \rho'(F) \cdot x \subset U.
  \]
  But then by compactness of $K$, we can find a \emph{single}
  neighborhood $\mc{W} \subset \vfg(\Gamma_p)$ so that
  $\rho'(F) \cdot K \subset U$, as required.
\end{proof}


\subsection{Extended convergence dynamics}

If $\Gamma$ is a relatively hyperbolic group, relative to a collection
$\mc{H}$ of peripheral subgroups, then $\Gamma$ acts as a
\emph{convergence group} on the Bowditch boundary $\bgamh$. We recall
the definition here.

\begin{definition}
  \label{defn:convergence_group}
  Let $\Gamma$ act on a topological space $M$. $\Gamma$ acts as a
  \emph{convergence group} if for every infinite sequence of distinct
  elements $\gamma_n \in \Gamma$, there exist points $a, b \in M$ and
  a subsequence $\gamma_m \in \Gamma$ such that $\gamma_m$ converges
  uniformly on compacts in $M - \{a\}$ to the constant map $b$.
\end{definition}

When $\gamma_n$ is a sequence of distinct elements in a relatively
hyperbolic group $\Gamma$, then $\gamma_n$ converges to $b$ uniformly
on compacts in $\bgamh - \{a\}$ if and only if
$\gamma_n$ converges to $b$ in the compactification
$\overline{\Gamma} = \Gamma \sqcup \bgamh$ and the
inverse sequence $\gamma_n^{-1}$ converges to $a$.

Recall that if a group $\Gamma$ acts by homeomorphisms on a Hausdorff
space $X$, the pair $(\Gamma, X)$ is called a \emph{topological
  dynamical system}. We say that an \emph{extension} of $(\Gamma, X)$
is a topological dynamical system $(\Gamma, Y)$ together with a
$\Gamma$-equivariant surjective map $\phi:Y \to X$.

In this paper, when $(\Gamma, \mc{H})$ is a relatively hyperbolic pair
(i.e. $\Gamma$ is hyperbolic relative to a collection $\mc{H}$ of
peripheral subgroups), we will consider \emph{embedded extensions} of
the topological dynamical system $(\Gamma, \bgamh)$. We want these
embedded extensions to preserve the convergence dynamics of
$(\Gamma, \bgamh)$, in the following sense:
\begin{restatable}{definition}{dynamicsPreserving}
  \label{defn:extends_convergence_dynamics}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, with
  $\Gamma$ acting on a connected compact metrizable space $M$ by
  homeomorphisms. Let $\Lambda \subset M$ be a closed
  $\Gamma$-invariant set.
  
  We say that a continuous equivariant surjective map
  $\phi:\Lambda \to \bgamh$ \emph{extends the convergence dynamics of
    $\Gamma$} if for each $z \in \bgamh$, there exists an open set
  $C_z \subset M$ containing $\Lambda - \phi^{-1}(z)$, satisfying the
  following:

  If $\gamma_n$ is a sequence in $\Gamma$ with
  $\gamma_n^{\pm 1} \to z_\pm$ for $z_\pm \in \bgamh$,
  then for any compact set $K \subset C_{z_-}$ and any open set $U$
  containing $\phi^{-1}(z_+)$, for sufficiently large $n$,
  $\gamma_n \cdot K$ lies in $U$.
\end{restatable}

Now let $G$ denote a semisimple Lie group with no compact factor. We
are interested in the action of relatively hyperbolic groups on
certain homogeneous spaces associated to $G$.

\begin{restatable}{definition}{EGF}
  \label{defn:boundary_dynamics}
  Let $\Gamma$ be a relatively hyperbolic group, let
  $\rho:\Gamma \to G$ be a representation, and let $P \subset G$ be a
  symmetric parabolic subgroup. We say that $\rho$ is \emph{extended
    geometrically finite} (EGF) with respect to $P$ if there exists a
  closed $\rho(\Gamma)$-invariant set $\Lambda \subset G/P$ and a
  continuous $\rho$-equivariant surjective antipodal map
  $\phi:\Lambda \to \bgamh$ extending the convergence dynamics of
  $\Gamma$.

  The map $\phi$ is called a \emph{boundary extension} of the
  representation $\rho$, and the closed invariant set $\Lambda$ is
  called the \emph{boundary set}.
\end{restatable}

We refer to \cref{sec:egf_representations} for the definition of
``antipodal map'' in this context.

Like (relative) Anosov representations, extended geometrically finite
representations are always discrete with finite kernel (see
\ref{sec:discrete_finite_kernel}). The central result of this paper is
that EGF representations have a \emph{relative stability} property: if
$\rho$ is an EGF representation, then any small deformation of $\rho$
which satisfies some conditions on the peripheral subgroups is also
EGF.

To state this theorem, we define a notion of a \emph{peripherally
  stable} subspace of $\Hom(\Gamma, G)$. The precise definition is
given in \cref{sec:relative_stability}, but roughly speaking, a
subspace $W \subseteq \Hom(\Gamma, G)$ is \emph{peripherally stable}
if the large-scale dynamical behavior of the peripheral subgroups of
$\Gamma$ is in some sense preserved by small deformations inside of
$W$. Notably, the action of a deformed peripheral subgroup does
\emph{not} need to be even topologically conjugate to the action of
the original peripheral subgroup. In particular, the number of
attracting fixed points of a peripheral subgroup can change under a
peripherally stable deformation.

We prove the following:
\begin{restatable}{theorem}{cuspStableStability}
  \label{thm:cusp_stable_stability}
  Let $\rho:\Gamma \to G$ be EGF with respect to $P$, let
  $\phi:\Lambda \to \bgamh$ be a boundary extension, and let
  $\mc{W} \subseteq \Hom(\Gamma, G)$ be peripherally stable at
  $(\rho, \phi)$. For any compact subset $Z$ of $\bgamh$ and any open
  set $V \subset \flags$ containing $\phi^{-1}(Z)$, there is an open
  subset $\mc{W}' \subset \mc{W}$ containing $\rho$ such that each
  $\rho' \in \mc{W}$ is EGF with respect to $P$, and has an EGF
  boundary extension $\phi'$ satisfying $\phi'^{-1}(Z) \subset V$.
\end{restatable}
Note that when $\rho:\Gamma \to G$ is a $P$-Anosov representation of a
hyperbolic group $\Gamma$, then the associated boundary embedding
$\dee \Gamma \to \flags$ also varies continuously with $\rho$ in the
compact-open topology on maps $\dee\Gamma \to \flags$. Since EGF
representations come with boundary \emph{extensions} (rather than
embeddings), \Cref{thm:cusp_stable_stability} only gives us a
\emph{semicontinuity} result.

While the peripheral stability condition in
\Cref{thm:cusp_stable_stability} is mildly technical, we can apply it
to yield more concrete results.
\begin{definition}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and let
  $\rho:\Gamma \to G$ be a representation. The space of
  \emph{cusp-preserving} representations
  \[
    \Hom_{\mr{cp}}(\Gamma, G, \mc{H}, \rho)
  \]
  is the set of representations $\rho':\Gamma \to G$ such that for
  each peripheral subgroup $H \in \mc{H}$, we have
  \[
    \rho'|_H = g \cdot \rho|_H \cdot g^{-1}
  \]
  for some $g \in G$ (which may depend on $H$).
\end{definition}

\begin{cor}
  \label{cor:cusp_preserving_stability}
  Let $\rho:\Gamma \to G$ be an EGF representation. Then there is a
  neighborhood of $\rho$ in $\Hom_{\mr{cp}}(\Gamma, G, \mc{H}, \rho)$
  consisting of EGF representations.
\end{cor}

\Cref{cor:cusp_preserving_stability} gives a very restrictive example
of a peripherally stable subspace of $\Hom(\Gamma, G)$. But, in
general the peripheral stability condition is flexible enough to allow
peripheral subgroups to deform in nontrivial ways.

In particular, it is possible to find peripherally stable deformations
of EGF representations which change the Jordan block decomposition of
elements in the peripheral subgroups. For instance, one can deform an
EGF representation in $\SL(d, \R)$ with unipotent peripheral
subgroups into an EGF representation with diagonalizable peripheral
subgroups---see \Cref{ex:peripheral_stability}.

\subsection{Relation to (relative) Anosov representations}
EGF representations generalize \emph{relative asymptotic
  embeddedness}, one of the definitions of relative Anosov
representation given by Kapovich-Leeb in \cite{kl2018relativizing}.

All of the Kapovich-Leeb definitions are given in terms of a
\emph{model face} $\tau_{\mr{mod}}$ of a \emph{model spherical Weyl
  chamber} $\sigma_{\mr{mod}}$ for the Riemannian symmetric space
$G/K$ (see \cref{sec:taumod_regular}). Such a face $\taumod$ uniquely
determines a conjugacy class of parabolic subgroup in $G$; we let
$P_\taumod$ denote any parabolic subgroup in this conjugacy class.

\begin{restatable}{theorem}{relAsympBdryDynamics}
  \label{thm:bdry_dynamics_homeo_equiv_rel_asymp_embedded}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair. A
  representation $\rho:\Gamma \to G$ is relatively
  $\tau_{\mr{mod}}$-asymptotically embedded if and only if $\rho$ is
  EGF with respect to $P_{\tau_{\mr{mod}}}$ and has a boundary
  extension $\phi:\Lambda \to \bgamh$ which is a
  homeomorphism.
\end{restatable}

An EGF representation $\rho$ has a boundary extension which is a
homeomorphism precisely when the images of the peripheral subgroups
have a unique limit point in the flag manifold $G/P$ (see
\Cref{prop:conical_pts_singletons}). So
\Cref{cor:cusp_preserving_stability} implies the following:
\begin{cor}
  \label{cor:asymp_embedded_rel_stable}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and let
  $\rho:\Gamma \to G$ be a relatively $\taumod$-asymptotically
  embedded representation. There is an open neighborhood of $\rho$ in
  $\Hom_{\mr{cp}}(\Gamma, G, \mc{H}, \rho)$ consisting of relatively
  $\taumod$-asymptotically embedded representations.
\end{cor}

\begin{remark}
  Kapovich and Leeb assert in \cite{kl2018relativizing} that
  relatively asymptotically embedded representations are stable under
  certain small relative deformations---possibly referring to a result
  along the lines of \Cref{cor:asymp_embedded_rel_stable}.

  In addition, in \cite{czz2021cusped}, Canary-Zhang-Zimmer study a
  notion of \emph{$P_k$-Anosov representations} for geometrically
  finite Fuchsian groups. These representations are always
  \emph{relatively dominated} in the sense of Zhu, which means they
  are also relatively asymptotically embedded by
  \cite{zhu2019relatively}, Theorem 9.4. Canary-Zhang-Zimmer prove
  that $P_k$-Anosov representations are stable in the space of
  \emph{type-preserving} deformations, which implies
  \Cref{cor:asymp_embedded_rel_stable} in this case.
\end{remark}

\Cref{thm:cusp_stable_stability} gives one potential way to find more
complicated discrete deformations of relatively
asymptotically-embedded representations. For instance, in
\cref{sec:generalized_cusps}, we give examples of deformations of
relatively asymptotically-embedded representations where the deformed
group is still EGF, but the peripheral subgroups do \emph{not} have
unique limit points in $G/P$.

\subsubsection{Hyperbolicity and Anosov representations}

When $\Gamma$ is a \emph{hyperbolic} group (i.e. if $\Gamma$ is
relatively hyperbolic relative to an empty collection of peripheral
subgroups), then the Bowditch boundary of $\Gamma$ is identified with
its Gromov boundary, and \emph{relative} $\taumod$-asymptotic
embeddedness is the same as the notion of $\taumod$-asymptotic
embeddedness defined in \cite{klp2017anosov}. A representation
$\rho:\Gamma \to G$ is $\taumod$-asymptotically embedded if and only
if it is $P_\taumod$-Anosov (see \cite{klp2017anosov}, Theorem
1.1). In particular, this implies:

\begin{cor}
  \label{cor:hyperbolic_egf_anosov}
  Let $\rho:\Gamma \to G$ be a representation of a hyperbolic
  group. Then $\rho$ is EGF with respect to $P$ if and only if $\rho$
  is $P$-Anosov.
\end{cor}

\subsubsection{Relativization}

When $\Gamma$ is a relatively hyperbolic group, the Bowditch boundary
of $\Gamma$ (and thus, the definition of an EGF representation)
depends on a choice of \emph{peripheral structure} for $\Gamma$,
i.e. a choice of collection of peripheral subgroups. In general, there
might be more than one possible choice: for instance, if a group
$\Gamma$ is hyperbolic relative to a collection $\mc{H}$ of hyperbolic
subgroups, then $\Gamma$ is itself hyperbolic, relative to an
\emph{empty} collection of peripheral subgroups (see
\cite{ds2005tree}, Corollary 1.14).

In this situation, we have the following \emph{Anosov relativization
  theorem}:
\begin{theorem}
  \label{thm:changing_peripheral_structure}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and suppose
  that each $H \in \mc{H}$ is hyperbolic. If $\rho:\Gamma \to G$ is an
  EGF representation with respect to $P$ for the peripheral structure
  $\mc{H}$, and $\rho$ restricts to a $P$-Anosov representation on
  each $H \in \mc{H}$, then $\rho$ is a $P$-Anosov representation of
  $\Gamma$.
\end{theorem}

\subsection{Relation to convex projective structures}

\subsubsection{Convex cocompactness in projective space}

In \cite{dgk2017convex}, Danciger-Gu\'eritaud-Kassel gave a definition
of a \emph{convex cocompact} projective orbifold, generalizing the
notion of a group acting convex cocompactly in hyperbolic space. Their
work (together with work of Zimmer \cite{zimmer2021projective}) shows
that there is a close connection between Anosov representations and
convex cocompact projective orbifolds with hyperbolic fundamental
group: the holonomy representation in $\PGL(d+1, \R)$ of such an
orbifold is always $P_1$-Anosov (where $P_1 \subset \PGL(d+1, \R)$ is
the parabolic subgroup stabilizing a line in $\R^{d+1}$). Moreover,
any Anosov representation can be composed with some irreducible
representation to obtain the holonomy of one of these projective
structures.

There are also examples of convex cocompact projective orbifolds with
\emph{relatively} hyperbolic fundamental group---see for instance
\cite{benoist2006convexes}, \cite{clm2020convex},
\cite{dgklm2021convex}. When the collection of peripheral subgroups is
not empty, these examples are \emph{never} relatively Anosov in the
sense of Kapovich-Leeb: Danciger-Gu\'eritaud-Kassel show that if
$\Gamma \subset \PGL(d+1, \R)$ is a subgroup acting convex cocompactly
in $\RP^d$, then $\Gamma$ contains no nontrivial unipotent elements,
and relatively asymptotically embedded representations always have
unipotent peripherals.

Nevertheless, our previous work \cite{weisman2020dynamical} (together
with subsequent work of Islam-Zimmer \cite{iz2022structure}) shows
that in this situation there is still a reasonable ``boundary map''
from $\bgamh$ into a quotient of projective space. This boundary map
also fits into the framework of EGF representations:

\begin{restatable}{theorem}{convexCocompactBoundaryDynamics}
  \label{thm:convex_cocompact_boundary_dynamics}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and suppose
  $\rho:\Gamma \to \PGL(d+1, \R)$ is convex cocompact in $\RP^d$. Then
  $\rho$ is EGF with respect to $\projstab$, where $\projstab$ is the
  stabilizer of a flag of type $(1, d)$ in $\R^{d+1}$.
 
 
 
\end{restatable}

In \cite{dgk2017convex}, Danciger-Gu\'eritaud-Kassel show (using work
of Cooper-Long-Tillmann \cite{clt2018deforming}) that in fact convex
cocompact representations are always \emph{absolutely} stable in
$\Hom(\Gamma, \PGL(d+1, \R))$. This tells us that if $\rho$ is a
projectively convex cocompact representation of a relatively
hyperbolic group $\Gamma$, then an open neighborhood of $\rho$ in
$\Hom(\Gamma, \PGL(d+1, \R))$ consists of EGF representations (without
any need to invoke \Cref{thm:cusp_stable_stability}).

\subsubsection{Geometrically finite strictly convex projective
  manifolds}

In \cite{cm2014finitude}, Crampon-Marquis define a notion of
geometrical finiteness for groups acting on strictly convex domains in
projective space. Zhu \cite{zhu2019relatively} showed that these
groups are always \emph{relatively dominated}, and that relatively
dominated representations are always relatively asymptotically
embedded. So an immediate corollary of Zhu's results and
\Cref{thm:bdry_dynamics_homeo_equiv_rel_asymp_embedded} is:
\begin{cor}
  \label{cor:cm_geo_finite_pcbd}
  Let $\Omega \subseteq \RP^d$ be a strictly convex domain, and
  suppose that $\Gamma \subseteq \Aut(\Omega)$ acts on $\Omega$
  geometrically finitely in the sense of Crampon-Marquis. Then the
  inclusion $\Gamma \hookrightarrow \PGL(d+1, \R)$ is EGF with respect
  to $P_{1,d}$.
\end{cor}

\subsubsection{Convex projective orbifolds with generalized cusps}

In \cite{clt2018deforming}, Cooper-Long-Tillmann studied relative
stability properties for holonomy representations of another class of
noncompact convex projective orbifolds---those with \emph{generalized
  cusps}. They consider the situation of a convex projective manifold
$M$ (with strictly convex boundary) which is a union of a compact
piece and finitely many ends homeomorphic to $N \times [0, \infty)$,
where $N$ is a compact manifold with virtually abelian fundamental
group. The ends of such a manifold are called generalized cusps.

In \cite{bcl2020generalized}, Ballas-Cooper-Leitner classified
generalized cusps in $\RP^d$ into $d + 1$ different \emph{cusp
  types}. A \emph{type 0 cusp} is projectively equivalent to a cusp in
$\H^d$ and has virtually unipotent holonomy, while a \emph{type $d$
  cusp} has virtually diagonalizable holonomy. The remaining cusp
types ``interpolate'' between these two.

If $M = \Omega / \Gamma$ is a finite-volume convex projective
$d$-manifold, and $\Omega \subseteq \RP^d$ is a strictly convex
domain, then work of Crampon-Marquis \cite{cm2014finitude} (see also
Cooper-Long-Tillmann \cite{clt2015convex}) shows that $M$ decomposes
into a union of a compact piece and finitely many type 0 cusps, and
$\pi_1M$ is hyperbolic relative to the collection of cusp groups
\[
  \mc{H} = \{\pi_1C_i : C_i \textrm{ a cusp in } M\}.
\]
\Cref{cor:cm_geo_finite_pcbd} implies that the holonomy
$\rho:\pi_1M \to \PGL(d+1, \R)$ of such a manifold is an EGF
representation, and \Cref{thm:cusp_stable_stability} says that
peripherally stable deformations of $\rho$ are also EGF.

In this situation, we can give an explicit example of a subspace of
$\Hom(\pi_1M, \PGL(d+1))$ which is peripherally stable. Following
Cooper-Long-Tillman, we let $\vfg_{d+1}$ denote the set of
\emph{virtual flag groups}: discrete subgroups of $\PGL(d+1, \R)$
which have a finite-index subgroup which is conjugate to a group of
upper-triangular matrices.

\begin{definition}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair. We let
  \[
    \vfgreps(\Gamma, \mc{H})
  \]
  denote the space
  \[
    \{\rho \in \Hom(\Gamma, \PGL(d+1, \R)) : \forall H_i \in \mc{H},
    \rho(H_i) \in \vfg(d+1) \}.
  \]
\end{definition}

Let $\rho:\pi_1M \to \PGL(d+1, \R)$ be the holonomy of the manifold
$M$. \cite[Theorem 0.1]{clt2018deforming} says that the image of the
holonomy of each cusp of $M$ lies in $\vfg(d+1)$---i.e.
$\rho \in \vfgreps(\pi_1M, \mc{H})$. And, if $\rho'$ is any small
deformation of $\rho$ in $\vfgreps(\pi_1M, \mc{H})$, then $\rho'$ is
the holonomy of a nearby convex projective structure on $M$.

It turns out that these nearby representations are also EGF:
\begin{restatable}{theorem}{finiteVolProjDefStable}
  \label{thm:finite_volume_projective_deformations_stable}
  Let $M = \Omega / \Gamma$ be a finite-volume convex projective
  manifold, and suppose that $\Omega$ is strictly convex (so that
  $\pi_1M$ is hyperbolic relative to the collection $\mc{H}$ of cusp
  groups and the holonomy $\rho:\pi_1M \to \PGL(d+1, \R)$ is
  $P_{1,d}$-EGF with boundary extension $\phi$).

  Then the space $\vfgreps(\pi_1M, \mc{H})$ is peripherally stable at
  $(\rho, \phi)$. In particular, due to
  \Cref{thm:cusp_stable_stability}, an open subset of
  $\vfgreps(\pi_1M, \mc{H})$ containing $\rho$ consists of EGF
  representations.
\end{restatable}

In \cite{bobb2019convex}, Bobb produced examples of convex projective
$d$-manifolds with cusps of any non-diagonalizable type for all
dimensions $d$. These examples are all small deformations of
finite-volume hyperbolic $d$-manifolds. So, a consequence of Bobb's
work and \Cref{thm:finite_volume_projective_deformations_stable} is
the following:

\begin{cor}
  In every dimension $d$ and for every $0 < t < d$, there exists a
  convex projective $d$-manifold with a type $t$ generalized cusp and
  EGF holonomy.
\end{cor}

\subsection{New examples: symmetric powers of convex cocompact
  representations in $\SL(V)$}

Using \Cref{thm:cusp_stable_stability}, we are also able to find new
examples of discrete representations of relatively hyperbolic groups
which are stable. The idea is as follows: suppose that $\Gamma$ is
hyperbolic relative to a nonempty collection of virtually abelian
subgroups with rank $\ge 2$, and let
$\overline{\rho}:\Gamma \to \PGL(V)$ be a convex cocompact
representation in the sense of Danciger-Gu\'eritaud-Kassel. Possibly
after replacing $\Gamma$ with a finite-index subgroup, we may lift
$\overline{\rho}$ to a representation $\rho:\Gamma \to \SL(V)$ with
discrete image and finite kernel; we will also refer to $\rho$ as a
\emph{convex cocompact} representation in $\P(V)$.

We can compose $\rho$ with the symmetric power
\[
  \tau_k:\SL(V) \to \SL(\mathrm{Sym}^kV)
\]
to obtain a new representation of $\Gamma$. We show the following:
\begin{theorem}
  \label{thm:symmetric_powers_boundary_dynamics}
  Let $\Gamma$ be hyperbolic relative to virtually abelian subgroups,
  and let $\rho:\Gamma \to \SL(V)$ be convex cocompact in $\P(V)$. For
  any $k \ge 1$, the representation $\tau_k \circ \rho$ is EGF.
\end{theorem}

In general, we do \emph{not} expect $\tau_k \circ \rho$ to act convex
cocompactly in projective space, so we cannot use
Danciger-Gu\'eritaud-Kassel's result to see that it is
stable. However, we will be able to apply
\Cref{thm:cusp_stable_stability} to see the following:

\begin{theorem}
  \label{thm:symmetric_powers_open}
  There is an open subset of $\tau_k \circ \rho$ in
  $\Hom(\Gamma, \SL(\mathrm{Sym}^kV))$ consisting of EGF
  representations.
\end{theorem}

\subsection{Acknowledgements}

The author thanks his advisor, Jeff Danciger, for encouragement and
many helpful conversations---without which this paper could not have
been written. The author also thanks Katie Mann and Jason Manning for
helping to simplify some of the arguments in
\Cref{sec:relative_automata,sec:extended_convergence}, and Daniel
Allcock for giving some useful advice on the approach in
\Cref{sec:symmetric_powers}. Further thanks are owed to Fanny Kassel,
Max Riestenberg, Feng Zhu, and Andy Zimmer for providing feedback on
various earlier versions of this project.

This work was supported in part by NSF grant DMS-1937215.



\subsection{Parabolic subgroups}

Let $K$ be a maximal compact subgroup of the semisimple Lie group $G$,
and let $X$ be the Riemannian symmetric space $G/K$. A subgroup
$P \subset G$ is a \emph{parabolic subgroup} if it is the stabilizer
of a point in the visual boundary $\dee_\infty X$ of $X$. Two
parabolic subgroups $P, Q$ are \emph{opposite} if there is a
bi-infinite geodesic $c$ in $X$ so that $P$ is the stabilizer of
$c(\infty)$ and $Q$ is the stabilizer of $c(-\infty)$.

The compact homogeneous $G$-space $\flags$ is called a \emph{flag
  manifold}. If $P$ and $Q$ are parabolic subgroups, then we say that
two flags $\xi^+ \in G/P$ and $\xi^- \in G/Q$ are \emph{opposite} if
the stabilizers of $\xi^+$, $\xi^-$ are opposite parabolic
subgroups. (In particular a conjugate of $Q$ must be opposite to $P$).

\subsection{Root space decomposition}

Let $\mf{g}$ be the Lie algebra of $G$, and let $\mf{k}$ be the Lie
algebra of the maximal compact $K$. We can decompose $\mf{g}$ as
$\mf{k} \oplus \mf{p}$, and fix a maximal abelian subalgebra
$\mf{a} \subset \mf{p}$ (the \emph{Cartan subalgebra}). The
restriction of the Killing form $B$ to $\mf{p}$ is positive definite,
so any Cartan subalgebra $\mf{a}$ is naturally endowed with a
Euclidean structure.

Since $\mf{a}$ is abelian, it acts semisimply on $\mf{g}$. So we let
$\Sigma \subset \mf{a}^*$ denote the set of \emph{roots} for this
Cartan subalgebra. We have a \emph{root space decomposition}
\[
  \mf{g} = \mf{g}_0 \oplus \bigoplus_{\alpha \in \Sigma}
  \mf{g}_\alpha,
\]
where $X \in \mf{a}$ acts on $\mf{g}_\alpha$ by multiplication by
$\alpha(X)$.

We choose a set of \emph{simple roots} $\Delta \subset \Sigma$ so that
each $\alpha \in \Sigma$ can be uniquely written as a linear
combination of elements of $\Delta$ with coefficients either all
nonnegative or all nonpositive. We let $\Sigma_+$ denote the
\emph{positive roots}, i.e. roots which are nonnegative linear
combinations of elements of $\Delta$.

The simple roots $\Delta$ determine a \emph{Euclidean Weyl chamber}
\[
  \mf{a}^+ = \{x \in \mf{a} : \alpha(x) \ge 0, \textrm{ for all }
  \alpha \in \Delta\}.
\]
The kernels of the roots $\alpha \in \Delta$ are the \emph{walls} of
the Euclidean Weyl chamber.

Choosing a maximal compact $K$, a Cartan subalgebra $\mf{a}$, and a
Euclidean Weyl chamber $\mf{a}^+$ determines a \emph{Cartan
  projection}
\[
  \mu:G \to \mf{a}^+,
\]
uniquely determined by the equation $g = k\exp(\mu(g))k'$, where
$k, k' \in K$ and $\mu(g) \in \mf{a}^+$.

\subsection{$P$-divergence}

Fix a subset $\theta$ of the simple roots $\Delta$. We define a
\emph{standard parabolic subgroup} $\stdpargroup$ to be the normalizer
of the Lie algebra
\[
  \bigoplus_{\alpha \in \Sigma^+_\theta} \mf{g}_\alpha,
\]
where $\Sigma^+_\theta$ is the set of positive roots which are
\emph{not} in the span of $\Delta - \theta$. The \emph{opposite
  subgroup} $\oppgroup$ is the normalizer of
\[
  \bigoplus_{\alpha \in \Sigma^+_\theta} \mf{g}_{-\alpha}.
\]

Every parabolic subgroup $P \subset G$ is conjugate to a unique
standard parabolic subgroup $\stdpargroup$, and every pair of opposite
parabolics $(P^+, P^-)$ is simultaneously conjugate to a unique pair
$(\stdpargroup, \stdoppgroup)$.

For a fixed $\theta \subset \Delta$, the group $\stdpargroup$ is the
stabilizer of the endpoint of a geodesic ray $\exp(t Z) \cdot p$,
where $p \in X$ is the image of the identity in $G/K$, and for any
$\alpha \in \Delta$, the element $Z \in \mf{a}^+$ satisfies
\[
  \alpha(Z) = 0 \iff \alpha \in \Delta - \theta.
\]

\begin{definition}
  Let $g_n$ be a sequence in $G$. The sequence $g_n$ is
  \emph{$\stdpargroup$-divergent} if for every $\alpha \in \theta$, we
  have
  \[
    \alpha(\mu(g_n)) \to \infty.
  \]

  That is, the Cartan projections of the sequence $g_n$ \emph{drift
    away from the walls} of $\mf{a}$ determined by the subset
  $\theta \subset \Delta$.

  For a general parabolic subgroup $P \subset G$, we say that $g_n$ is
  \emph{$P$-divergent} if $g_n$ is $\stdpargroup$-divergent for
  $\stdpargroup$ conjugate to $P$.
\end{definition}

\subsection{$\taumod$-regularity}
\label{sec:taumod_regular}

$P$-divergent sequences are equivalent to the \emph{$\taumod$-regular}
sequences discussed in the work of Kapovich-Leeb-Porti, where
$\taumod$ is the unique face corresponding to $P$ in a \emph{spherical
  model Weyl chamber}. We explain the connection here.

For any point $p \in X$, we let $\mf{p}$ be the uniquely determined
subspace of $\mf{g}$ such that $\mf{g} = \mf{k} \oplus \mf{p}$, where
$\mf{k}$ is the Lie algebra of the stabilizer of $p$ in $G$.

Let $z \in \dee_\infty X$. There is a point $p \in X$, a maximal
abelian subalgebra $\mf{a} \subset \mf{p}$, a Euclidean Weyl chamber
$\mf{a}^+ \subset \mf{a}$, and a unit-length $Z \in \mf{a}^+$ such
that $z$ is the endpoint of the geodesic ray
$c(t) = \exp(tZ) \cdot p$.

Up to the action of the stabilizer of $z$, the point $p$, the Cartan
subalgebra $\mf{a}$, the Euclidean Weyl chamber $\mf{a}^+$, and the
unit vector $Z \in \mf{a}^+$ are uniquely determined. In addition, the
stabilizer in $G$ of the triple $(p, \mf{a}, \mf{a}^+)$ acts trivially
on $\mf{a}^+$.

This means that we can identify the space $\dee_\infty X / G$ with the
set of unit vectors in any Euclidean Weyl chamber $\mf{a}^+$. This set
has the structure of a \emph{spherical simplex}. We let $\sigmamod$
denote the \emph{model spherical Weyl chamber} $\dee_\infty X / G$.

We let $\pi:\dee_\infty X \to \sigmamod$ be the \emph{type map} to the
model spherical Weyl chamber. For fixed $z \in \dee_\infty X$, we let
$P_z$ denote the parabolic subgroup stabilizing $z$.

After choosing a maximal compact $K$, a Cartan subalgebra $\mf{a}$,
and a Euclidean Weyl chamber $\mf{a}^+$, the data of a \emph{face}
$\taumod$ of the spherical simplex $\sigmamod$ is the same as the data
of a \emph{subset} of the simple roots of $G$: the set of roots
identifies a collection of walls of the Euclidean Weyl chamber
$\mf{a}^+$. The intersection of those walls with the unit sphere in
$\mf{a}$ is uniquely identified with a face of $\sigmamod$.

\begin{definition}
  Let $\taumod$ be a face of the model spherical Weyl chamber
  $\sigmamod$. We say that a sequence $g_n \in G$ is
  \emph{$\taumod$-regular} if $g_n$ is $P_z$-divergent for some
  $z \in \dee_\infty X$ such that $\pi(z) \in \taumod$.
\end{definition}

For a fixed model face $\taumod \subset \sigmamod$, we let $P_\taumod$
denote any parabolic subgroup which is the stabilizer of a point
$z \in \pi^{-1}(\taumod)$. All such parabolic subgroups are conjugate,
so as a $G$-space the flag manifold $G/P_\taumod$ depends only on the
model face $\taumod$.

\subsection{Affine charts}

\begin{definition}
  \label{defn:affine_chart}
  Let $\pargroup$, $\oppgroup$ be opposite parabolic subgroups in
  $G$. Given a flag $\xi \in \oppflags$, we define
  \[
    \Opp(\xi) = \{\eta \in \pflags : \xi \textrm{ is opposite to }
    \eta\}.
  \]
  We call a set of the form $\Opp(\xi)$ for some $\xi \in \oppflags$
  an \emph{affine chart} in $\pflags$.
\end{definition}

An affine chart is the unique open dense orbit of $\Stab_G(\xi)$ in
$\pflags$. When $G = \PGL(d+1, \R)$ and $\pargroup$ is the stabilizer
of a line $\ell \subset \R^{d+1}$, $\pflags$ is identified with
$\RP^d$ and this notion of affine chart agrees with the usual one in
$\RP^d$.

\subsection{Dynamics in flag manifolds}
\label{sec:flag_dynamics_background}

There is a close connection between $P$-divergence in the group $G$
and the \emph{topological dynamics} of the action of $G$ on the
associated flag manifold $G/P$. Kapovich-Leeb-Porti frame this
connection in terms of a \emph{contraction property} for $P$-divergent
sequences.

\begin{definition}[\cite{klp2017anosov}, Definition 4.1]
  Let $g_n$ be a sequence of group elements in $G$. We say that $g_n$
  is \emph{$\pargroup$-contracting} if there exist $\xi \in \pflags$,
  $\xi_- \in \oppflags$ such that $g_n$ converges uniformly to $\xi$
  on compact subsets of $\Opp(\xi_-)$.

  The flag $\xi$ is the uniquely determined \emph{limit} of the
  sequence $g_n$.
\end{definition}

\begin{definition}
  \label{defn:limit_set}
  For an arbitrary sequence $g_n \in G$, a \emph{$\pargroup$-limit
    point} of $g_n$ in $\pflags$ is the limit point of some
  $\pargroup$-contracting subsequence of $g_n$.

  The \emph{$\pargroup$-limit set} of a group $\Gamma \subset G$ is the
  set of $\pargroup$-limit points of sequences in $\Gamma$.
\end{definition}

The importance of contracting sequences is captured by the following:
\begin{prop}[\cite{klp2017anosov}, Proposition 4.15]
  \label{prop:divergence_equivalent_contracting}
  A sequence $g_n \in G$ is $P^+$-divergent if and only if every
  subsequence of $g_n$ has a $P^+$-contracting subsequence.
\end{prop}

\Cref{prop:divergence_equivalent_contracting} implies in particular
that if $g_n \in G$ is $\pargroup$-divergent, then up to subsequence
there is an open subset $U \subset \pflags$ such that $g_n \cdot U$
converges to a singleton in $\pflags$. It turns out that this ``weak
contraction property'' is enough to characterize $P^+$-divergence.

\begin{restatable}{prop}{localContractingImpliesGlobalContracting}
  \label{prop:local_contracting_implies_global_contracting}
  Let $g_n$ be a sequence in $G$, and suppose that for some nonempty
  open subset $U \subset \pflags$, we have $g_n \cdot U \to \{\xi\}$
  for $\xi \in \pflags$. Then $g_n$ is $\pargroup$-divergent, and has
  a unique $\pargroup$-limit point $\xi \in \pflags$.
\end{restatable}

We provide a proof of this fact in \Cref{sec:contraction_appendix}.

\subsubsection{Dynamics of inverses of $\pargroup$-divergent
  sequences}

When $g_n$ is a $\pargroup$-divergent sequence, the inverse sequence
is $P^-$-divergent. Kapovich-Leeb-Porti show that this can be framed
in terms of the dynamical behavior of the inverse sequence.
\begin{lem}[\cite{klp2017anosov}, Lemma 4.19]
  \label{lem:klp_contraction_symmetry}
  For $g_n \in G$ and flags $\xi_- \in \oppflags, \xi_+ \in \pflags$,
  the following are equivalent:
  \begin{enumerate}
  \item $g_n$ is $P^+$-contracting and $g_n|_{\Opp(\xi_-)} \to \xi_+$
    uniformly on compacts.
  \item $g_n$ is $P^+$-divergent, $g_n$ has unique $P^+$-limit point
    $\xi_+$, and $g_n^{-1}$ has unique $P^-$-limit point $\xi_-$.
  \end{enumerate}
\end{lem}






\section{Introduction}

\subfile{introduction.tex}

\part{Extended geometrically finite representations}
\label{part:egfreps}

The first part of this paper introduces extended geometrically finite
representations. After providing some background in
\Cref{sec:rel_hyp_background,sec:lie_theory_background}, we give the
formal definition of EGF representations (and explain their connection
to Anosov representations) in \Cref{sec:egf_definition}.

The rest of this part of the paper is devoted to the proof of the
relative stability theorem for extended geometrically finite
representations (\Cref{thm:cusp_stable_stability}). In
\Cref{sec:relative_automata} and \Cref{sec:extended_convergence}, we
introduce the main technical tool we need for the proof: a
\emph{relative quasigeodesic automaton} associated to the action of
any non-elementary relatively hyperbolic group on its Bowditch
boundary. In \Cref{sec:contracting_paths}, we relate relative
quasigeodesic automata to the action of a relatively hyperbolic group
on a flag manifold. We use these tools to develop an alternative
characterization of EGF representations in \Cref{sec:weak_egf}, and
finally prove our main theorem in \Cref{sec:relative_stability}.

The general approach to our proof of \Cref{thm:cusp_stable_stability}
is loosely inspired by Sullivan's proof of stability for convex
cocompact groups in rank-one
\cite{sullivan1985quasiconformal}. Sullivan shows that a discrete
group $\Gamma \subset \PO(d,1)$ is convex cocompact if and only if the
action of $\Gamma$ on its limit set $\Lambda$ in $\dee \H^d$ satisfies
an \emph{expansion} property. Then, he uses this expansion property to
give a \emph{symbolic coding} for infinite quasigeodesic rays in
$\Gamma$. This coding gives a way to see that the correspondence
between geodesic rays in $\Gamma$ and points in $\dee \H^d$ is stable
under small perturbations of the representation. Notably, in
\cite{kkl2019structural}, Kapovich-Kim-Lee follow Sullivan's basic
procedure to provide a detailed proof of his result in a much more
general setting, in particular showing stability properties of uniform
lattices in higher-rank Lie groups. The ideas in that paper guide some
of our approach below.

In \cite{bps2019anosov}, Bochi-Potrie-Sambarino used a related
technique to give a proof of the fact that $P_k$-Anosov
representations of hyperbolic groups into $\PGL(d+1, \R)$ are
stable. They consider the \emph{geodesic automaton} for a hyperbolic
group $\Gamma$. This is a finite directed graph $\mc{G}$ whose edges
are labelled with generators of $\Gamma$. Bochi-Potrie-Sambarino show
that a representation $\rho:\Gamma \to \PGL(d+1, \R)$ is $P_k$-Anosov
if and only if there is a way to assign an open subset $U_v$ of
$\RP^d$ to each vertex $v$ of $\mc{G}$ so that each edge $(a,b)$ of
$\mc{G}$ corresponds to an inclusion
\begin{equation}
  \label{eq:automaton_inclusions}
  \rho(\gamma_{a,b}) \cdot \overline{U_b} \subset U_a.
\end{equation}
This shows that it is possible to verify if a small deformation of an
Anosov representation is Anosov by checking a finite set of open
conditions.

In a sense, this approach can be thought of as a kind of
\emph{generalized ping-pong argument} for non-free groups. Indeed,
when $\Gamma$ is actually a free group, the inclusions in
\eqref{eq:automaton_inclusions} are exactly the inclusions of sets
needed to set up ping-pong.

We take a somewhat similar approach to prove our relative stability
theorem. Given an extended geometrically finite representation $\rho$,
we construct a \emph{relative quasigeodesic automaton} $\mc{G}$, and
open subsets $U_v \subset G/P$ for each vertex $v$ of $\mc{G}$,
satisfying certain inclusions corresponding to the edges of
$\mc{G}$. We show that if such a system of open subsets also exists
for a nearby representation $\rho'$, then $\rho'$ must also be
extended geometrically finite.

It is helpful to keep an important special case in mind. If $A, B$ are
finitely generated groups, then $A * B$ is hyperbolic relative to the
collection of conjugates of $A$ and $B$. In this case, the inclusions
of sets determined by a relative quasigeodesic automaton are nothing more
than the inclusions of sets required to set up a ping-pong argument
proving that a representation of $A * B$ is discrete and faithful.

For a general relatively hyperbolic group $\Gamma$, we essentially use
the convergence dynamics of $\Gamma$ to encode points in $\bgamh$
using infinite paths in the graph $\mc{G}$. This is closely related to
the idea of describing the \emph{geodesic flow} of the group with
symbolic dynamics; see for instance \cite{series1981symbolic}.

Our approach only yields a \emph{relative} stability result because,
unlike in the non-relative case, some of the edges of the relative
automaton $\mc{G}$ correspond to an infinite number of inclusions of
open sets. The proof shows that if $\rho$ is EGF, any deformation
$\rho'$ which respects the conditions imposed by these ``parabolic
edges'' must also be EGF.

\section{Relative hyperbolicity}
\label{sec:rel_hyp_background}

\subfile{relhyp_background.tex}

\section{Lie theory notation and background}
\label{sec:lie_theory_background}

\subfile{lie_theory.tex}

\section{EGF representations and relative Anosov representations}
\label{sec:egf_definition}
\subfile{egf_representations.tex}

\section{Relative quasigeodesic automata}
\label{sec:relative_automata}
\subfile{automata.tex}

\section{Extended convergence dynamics}
\label{sec:extended_convergence}
\subfile{extended_convergence.tex}

\section{Contracting paths in flag manifolds}
\label{sec:contracting_paths}
\subfile{flag_contraction.tex}

\section{A weaker criterion for EGF representations}
\label{sec:weak_egf}

\subfile{weak_egf.tex}

\section{Relative stability}
\label{sec:relative_stability}
\subfile{stability.tex}

\part{Examples}
\label{part:examples}

The rest of the paper is devoted to examples of EGF
representations. All of the examples we provide derive from
\emph{convex real projective structures} on manifolds.

\begin{notation}
  We establish the rest of the following conventions for the rest of
  the paper.

  \begin{itemize}
  \item If $V$ is a real vector space, then $\P(V)$ is projective
    space $(V - \{0\})/\sim$, where $u \sim v$ if $u = \lambda v$ for
    $\lambda \in \R$.

  \item For any $x \in V - \{0\}$, we let $[x]$ denote the image of
    $x$ under the quotient map $V - \{0\} \to \P(V)$.

  \item If $W$ is a subset of $V$, then we let $[W]$ denote the image
    of $W - \{0\}$ in $\P(V)$. If $W \subseteq V$ is a vector
    subspace, we will identify $\P(W)$ with $[W] \subset \P(V)$.

  \item When $B \subset \P(V)$, then the \emph{span} of $B$, denoted
    $\spn(B)$, is the subspace of $V$ spanned by any lift of $B$ in
    $V$. The \emph{projective span} of $B$ is the subset
    $[\spn(B)] \subset \P(V)$.
  \item If $w$ is any element of the \emph{dual} projective space
    $\P(V^*)$, we let $\hyp{w}$ denote the image of $\ker{\tilde{w}}$
    in $\P(V)$, where $\tilde{w}$ is any lift of $w$ in $V^*$.
  \item If $w \in \P(V^*)$ and $v \in \P(V)$ satisfy
    $v \notin \hyp{w}$, then we say $w$ and $v$ are \emph{transverse}
    and write $w \perp v$.
  \item We let $\projflags(V)$ denote the space of partial flags
    $(V_-, V_+)$, where $V_- \subset V_+ \subset V$, and
    $\dim(V_-) = \codim(V_+) = 1$. If $\dim(V) = d + 1$, then we say
    $\projflags(V)$ is the space of flags of \emph{type $(1, d)$}. If
    $V$ is understood from context we will sometimes just write
    $\projflags = \projflags(V)$.
  \end{itemize}
\end{notation}

\section{Convex cocompact representations in $\RP^d$}
\label{sec:convex_projective_egf}

\subfile{convex_projective.tex}

\section{Generalized cusps}

\subfile{generalized_cusps.tex}

\section{Symmetric powers of convex cocompact groups}

\subfile{symmetric_powers.tex}



\subsection{Geometrically finite actions}

Recall that a finitely generated group $\Gamma$ is \emph{hyperbolic}
(or \emph{word-hyperbolic} or \emph{$\delta$-hyperbolic} or
\emph{Gromov-hyperbolic}) if and only if it acts properly
discontinuously and cocompactly on a $\delta$-hyperbolic proper
geodesic metric space $Y$.

A \emph{relatively hyperbolic group} is also a group with an action by
isometries on a $\delta$-hyperbolic proper geodesic metric space $Y$,
but instead of asking for the action to cocompact, we ask for the
action to be in some sense ``geometrically finite.''

To be precise, this means that $Y$ has a $\Gamma$-invariant
decomposition into a \emph{thick part} $Y_{\mr{th}}$ and a countable
collection $\mc{B}$ of \emph{horoballs}. For a horoball $B$, we let
$\ctr(B)$ denote the center of $B$ in $\dee Y$, and we let $\Gamma_p$
denote the stabilizer of any $p \in \dee Y$.

\begin{definition}
  \label{defn:relatively_hyperbolic_group}
  Let $\Gamma$ be a finitely generated group acting on a hyperbolic
  metric space $Y$, and let $\mc{B}$ be a countable collection of
  horoballs in $Y$, invariant under the action of $\Gamma$ on $Y$. If:
  \begin{enumerate}
  \item The action of $\Gamma$ on the closure of
    $Y_{\mr{th}} = Y - \bigcup_{B \in \mc{B}} B$ is cocompact, and
  \item for each $B \in \mc{B}$, the stabilizer of $\ctr(B)$ in
    $\Gamma$ is finitely generated and infinite,
  \end{enumerate}
  then we say that $\Gamma$ is a \emph{relatively hyperbolic group},
  relative to the collection
  $\mc{H} = \{\Stab_\Gamma(p) : p = \ctr(B) \textrm{ for } B \in
  \mc{B}\}$.
\end{definition}

\begin{definition}
  Let $\Gamma$ be a relatively hyperbolic group, relative to a
  collection of subgroups $\mc{H}$.
  \begin{itemize}
  \item The centers of the horoballs in $\mc{B}$ are called
    \emph{parabolic points} for the $\Gamma$-action on $\dee Y$. The
    set of parabolic points in $\dee Y$ is denoted $\parbdry Y$.
  \item The parabolic point stablizers
    $\mc{H} = \{\Stab_\Gamma(p) : p \in \parbdry Y\}$ are called
    \emph{peripheral subgroups}. We often write $\Gamma_p$ for
    $\Stab_\Gamma(p)$.
  \end{itemize}
\end{definition}

A group $\Gamma$ might be hyperbolic relative to different collections
$\mc{H}$, $\mc{H}'$ of peripheral subgroups. The collection $\mc{H}$
of peripheral subgroups is sometimes called a \emph{peripheral
  structure} for $\Gamma$.

\begin{definition}
  Let $\Gamma$ be a finitely generated group, and let $\mc{H}$ be a
  collection of subgroups. We say that $(\Gamma, \mc{H})$ is a
  \emph{relatively hyperbolic pair} if $\Gamma$ is hyperbolic relative
  to $\mc{H}$.
\end{definition}

\subsection{The Bowditch boundary}

\begin{definition}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, so that
  $\mc{H}$ is the set of stabilizers of parabolic points for an action
  of $\Gamma$ on a metric space $Y$ as in
  \Cref{defn:relatively_hyperbolic_group}. We say that $Y$ is a
  \emph{Gromov model} for the pair $(\Gamma, \mc{H})$.
\end{definition}

In general there is \emph{not} a unique choice of Gromov model for a
given relatively hyperbolic pair $(\Gamma, \mc{H})$, even up to
quasi-isometry. There are various ``canonical'' constructions for a
preferred quasi-isometry class of Gromov model, with certain desirable
metric properties (see e.g. \cite{bowditch2012relatively},
\cite{gm2008dehn}).

Given \emph{any} two Gromov models $Y$, $Y'$ for $(\Gamma, \mc{H})$,
there is always a $\Gamma$-equivariant homeomorphism
$\dee Y \to \dee Y'$ \cite{bowditch2012relatively}. The $\Gamma$-space
$\dee Y$ is the \emph{Bowditch boundary} of $(\Gamma, \mc{H})$. We
will denote it by $\bgamh$, or sometimes just
$\dee \Gamma$ when the collection of peripheral subgroups is
understood from context. Since a Gromov model $Y$ is a proper
hyperbolic metric space, $\bgamh$ is always compact and
metrizable.

\begin{definition}
  We say a relatively hyperbolic pair $(\Gamma, \mc{H})$ is
  \emph{elementary} if $\Gamma$ is finite or virtually cyclic, or if
  $\mc{H} = \{\Gamma\}$.
\end{definition}

Whenever $(\Gamma, \mc{H})$ is nonelementary, its Bowditch boundary
contains at least three points. The convergence properties of the
action of $\Gamma$ on $\bgamh$ (see below) imply that in
this case, $\bgamh$ is \emph{perfect} (i.e. contains no
isolated points).

\subsubsection{Cocompactness on pairs}

Let $Y$ be a Gromov model for a relatively hyperbolic pair
$(\Gamma, \mc{H})$. Since $Y$ is hyperbolic, proper, and geodesic, for
any compact subset $K \subset Y$, the space of bi-infinite geodesics
passing through $K$ is compact.

Given any distinct pair of points $u, v \in \dee Y$, there is a
bi-infinite geodesic $c$ in $Y$ joining $u$ to $v$. This geodesic must
pass through the thick part $Y_{\mathrm{th}}$ of $Y$, so up to the
action of $\Gamma$ it passes through a fixed compact subset
$K \subset Y_{\mathrm{th}}$.

This implies:
\begin{prop}
  \label{prop:pairs_cocompact}
  The action of $\Gamma$ on the space of distinct pairs in
  $\bgamh$ is cocompact.
\end{prop}

\subsection{Convergence group actions}
\label{subsec:convergence_groups}

If a group $\Gamma$ acts on a proper geodesic hyperbolic metric space
$Y$, we can characterize the geometrical finiteness of the action
entirely in terms of the topological dynamics of the action on
$\dee Y$. In particular, we can understand geometrical finiteness by
studying properties of \emph{convergence group actions}. See
\cite{tukia1994convergence}, \cite{tukia1998conical},
\cite{bowditch1999convergence} for further detail on such actions, and
justifications for the results stated in this section.

\begin{definition}
  \label{defn:convergence_group_properties}
  Let $\Gamma$ act as a convergence group (see
  \Cref{defn:convergence_group}) on a topological space $Z$.
  \begin{enumerate}
  \item A point $z \in Z$ is a \emph{conical limit point} if there
    exists a sequence $\gamma_n \in \Gamma$ and distinct points $a, b
    \in Z$ such that $\gamma_n z \to a$ and $\gamma_n y \to b$ for any
    $y \ne z$.
  \item An infinite subgroup $H$ is a \emph{parabolic subgroup} if it
    fixes a point $p \in Z$, and every infinite-order element of $H$
    fixes exactly one point in $Z$.
  \item A point $p \in Z$ is a \emph{parabolic point} if it is the
    fixed point of a parabolic subgroup.
  \item A parabolic point $p$ is \emph{bounded} if its stabilizer
    $\Gamma_p$ acts cocompactly on $Z - \{p\}$.
  \end{enumerate}
\end{definition}

The name ``conical limit point'' makes more sense in the context of
convergence group actions on boundaries of hyperbolic metric spaces.

\begin{definition}
  \label{defn:geometric_conical_limit}
  Let $Y$ be a hyperbolic metric space, and let $z \in \dee Y$. We say
  that a sequence $y_n \in Y$ \emph{limits conically to $z$} if there
  is a geodesic ray $c:\R^+ \to Y$ limiting to $z$ and a constant
  $D > 0$ such that
  \[
    d_Y(y_n, c(t_n)) < D
  \]
  for some sequence $t_n \to \infty$.
\end{definition}

A bounded neighborhood of a geodesic in a hyperbolic metric space
looks like a ``cone,'' hence ``conical limit.''

\begin{prop}[\cite{tukia1994convergence}, \cite{tukia1998conical}]
  \label{prop:conical_limit_point_interp}
  Let $\Gamma$ be a group acting properly discontinuously by
  isometries on a proper geodesic hyperbolic metric space $Y$, and fix
  a basepoint $y_0 \in Y$.

  Then $\Gamma$ acts on $\dee Y$ as a convergence group. Moreover, a
  point $z \in \dee Y$ is a conical limit point (in the dynamical
  sense of \Cref{defn:convergence_group_properties}) if and only if
  there is a sequence $\gamma_n \cdot y_0$ limiting conically to $z$
  (in the geometric sense of \Cref{defn:geometric_conical_limit}). In
  this case, there are distinct points $a, b \in \dee Y$ such that
  $\gamma_n^{-1} \cdot z \to a$ and $\gamma_n^{-1} z' \to b$ for any
  $z' \ne z$ in $\dee Y$.
\end{prop}

If $\gamma_n \cdot y_0$ limits conically to a point $z \in \dee Y$ for
some (hence any) basepoint $y_0 \in Y$, we just say that $\gamma_n$
limits conically to $z$.

\begin{theorem}[\cite{bowditch2012relatively}]
  Let $\Gamma$ be a group acting by isometries on a hyperbolic metric
  space $Y$. Then $\Gamma$ is a relatively hyperbolic group, acting on
  $Y$ as in \Cref{defn:relatively_hyperbolic_group}, if and only if:
  \begin{enumerate}
  \item The induced action of $\Gamma$ on $\dee Y$ is a convergence
    group action.
  \item Every point $z \in \dee Y$ is either a conical limit point or
    a bounded parabolic point.
  \end{enumerate}
\end{theorem}

Whenever a group $\Gamma$ acts as a convergence group on a perfect
compact metrizable space $Z$, every point in $Z$ is either a conical
limit point or a bounded parabolic point, and the set of orbits of
parabolic points is finite, we say the $\Gamma$-action on $Z$ is
\emph{geometrically finite}. This is justified by a theorem of Yaman
\cite{yaman2004topological}, which says that any such group action is
induced by the action of a relatively hyperbolic group on a Gromov
model $Y$ whose boundary is equivariantly homeomorphic to $Z$. We can
then identify the space $Z$ with the Bowditch boundary $\bgamh$. The
set of parabolic points in $Z$ coincides exactly with the set of fixed
points of peripheral subgroups.

\begin{definition}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair. We write
  \[
    \bgamh = \conbdry(\Gamma, \mc{H}) \sqcup \parbdry
    (\Gamma, \mc{H}),
  \]
  where $\conbdry(\Gamma, \mc{H})$ and $\parbdry(\Gamma, \mc{H})$ are
  respectively the conical limit points and parabolic points in
  $\bgamh$.
\end{definition}

\subsubsection{Compactification of $\Gamma$ and divergent sequences}

When $(\Gamma, \mc{H})$ is a relatively hyperbolic pair, there is a
natural topology on the set
\[
  \overline{\Gamma} = \Gamma \sqcup \bgamh
\]
making it into a \emph{compactification} of $\Gamma$
(i.e. $\overline{\Gamma}$ is compact, $\bgamh$ and
$\Gamma$ are both embedded in $\overline{\Gamma}$, and $\Gamma$ is an
open dense subset of $\overline{\Gamma}$).

\begin{definition}
  A sequence $\gamma_n \in \Gamma$ is \emph{divergent} if it leaves
  every bounded subset of $\Gamma$ (equivalently, if a subsequence of
  it consists of pairwise distinct elements).
\end{definition}
Up to subsequence, a divergent sequence $\gamma_n \in \Gamma$
converges to a point $z \in \bgamh$.

\subsection{The coned-off Cayley graph}

Whenever $(\Gamma, \mc{H})$ is a relatively hyperbolic pair, there are
only finitely many conjugacy classes of groups in $\mc{H}$. We can fix
a finite set $\mc{P}$ of conjugacy representatives for the groups in
$\mc{H}$. The set $\mc{P}$ corresponds to a finite set
$\Pi \subset \parbdry \Gamma$ of parabolic points, such that
\[
  \mc{P} = \{\Gamma_p : p \in \Pi\}.
\]
$\Pi$ contains exactly one point in each $\Gamma$-orbit in
$\parbdry\Gamma$.

\begin{definition}
  \label{defn:conedgraph}
  Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair, and fix a
  finite generating set $S$ for $\Gamma$ and finite collection of
  conjugacy representatives $\mc{P}$ for $\mc{H}$.

  The \emph{coned-off Cayley graph} $\conedgraph$ is a metric space
  obtained from the Cayley graph $\cay(\Gamma, S)$ as follows: for
  each coset $gP_i$ for $P_i \in \mc{P}$, we add a vertex
  $v(gP_i)$. Then, we add an edge of length 1 from each $h \in gP_i$
  to $v(gP_i)$.
\end{definition}

The quasi-isometry class of $\conedgraph$ is independent of the choice
of generating set $S$. When $(\Gamma, \mc{H})$ is a relatively
hyperbolic pair, $\conedgraph$ is a hyperbolic metric space. It is
\emph{not} a proper metric space if $\mc{H}$ is nonempty. The Gromov
boundary of $\conedgraph$ is equivariantly homeomorphic to the set
$\conbdry \Gamma$ of conical limit points in $\bgamh$.


\subsection{Deformations of EGF representations}

In general, the set of EGF representations is not an open subset of
$\Hom(\Gamma, G)$. However, it is relatively open in a subspace of
$\Hom(\Gamma, G)$ where we restrict the deformations of the peripheral
subgroups appropriately. Roughly speaking, we want to consider
subspaces $\mc{W} \subset \Hom(\Gamma, G)$ where the \emph{dynamical}
behavior of the peripheral subgroups changes continuously under
deformation. That is, if $\rho_t$ is a small deformation of a
representation $\rho_0$, where $\rho_0(\Gamma_p)$ attracts points
towards $\Lambda_p$ at a particular ``speed,'' then we want
$\rho_t(\Gamma_p)$ to attract points towards a small deformation of
$\Lambda_p$ at a similar ``speed.''

The precise condition is the following:

\begin{definition}
  \label{defn:peripheral_stability}
  Let $\rho_0:\Gamma \to G$ be an EGF representation with boundary
  extension $\phi:\Lambda \to \bgamh$, and let
  $\mc{W} \subset \Hom(\Gamma, G)$ contain $\rho_0$.

  We say that $\mc{W}$ is \emph{peripherally stable at
    $(\rho_0, \phi)$} if for every $p \in \parbdry(\Gamma, \mc{H})$,
  every open set $U$ containing $\phi^{-1}(p)$, every compact set
  $K \subset C_p$, and every cofinite set $T \subset \Gamma_p$ such
  that
  \[
    \rho_0(T) \cdot K \subset U,
  \]
  there is an open set $\mc{W}' \subset \mc{W}$ containing $\rho_0$,
  such that for every $\rho' \in \mc{W}'$, we have
  \[
    \rho'(T) \cdot K \subset U.
  \]
\end{definition}

We restate the main result of this section here:

\cuspStableStability*

\begin{remark}
  In \cite{bowditch1998spaces}, Bowditch explored the deformation
  spaces of geometrically finite groups $\Gamma \subset \PO(d,1)$, and
  gave an explicit discription of semialgebraic subspaces of
  $\Hom(\Gamma, \PO(d,1))$ in which small deformations of $\Gamma$ are
  still geometrically finite.

  Bowditch's deformation spaces are peripherally stable, so it seems
  desirable to find a general algebraic description of peripherally
  stable subspaces.

  Even in $\PO(d,1)$, the question is subtle, however. Bowditch also
  gives examples of geometrically finite representations
  $\rho:\Gamma \to \PO(d,1)$ (for $d \ge 4$) and deformations $\rho_t$
  of $\rho$ in $\Hom(\Gamma, \PO(d,1))$ such that the restriction of
  $\rho_t$ to each cusp group in $\Gamma$ is discrete, faithful, and
  parabolic, but $\rho_t$ is not even discrete; further examples exist
  where the deformation is discrete, but not geometrically finite.
\end{remark}

\begin{example}
  \label{ex:peripheral_stability}
  Let $B \in \SL(d, \R)$ be a $d$-dimensional Jordan block with
  eigenvalue 1 and eigenvector $v$, and let $A \in \SL(d + 2, \R)$ be
  the block matrix
  $\begin{psmallmatrix}B \\& 1\\ & & 1\end{psmallmatrix}$.

  Although $[v]$ is not quite an attracting fixed point of $A$, it is
  still an ``attracting subspace'' in the sense that if $K$ is any
  compact subset of $\RP^{d+1}$ which does not intersect a fixed
  hyperplane of $A$, then $A^n \cdot K$ converges to $\{[v]\}$ (see
  \Cref{prop:attracting_repelling_subspaces}). Via a ping-pong
  argument, one can use this ``attracting'' behavior to show that for
  some $k \ge 1$ and some $M \in \SL(d + 2, \R)$, the group $\Gamma$
  generated by $\alpha = A^k$ and $\beta = MA^kM^{-1}$ is a discrete
  free group with free generators $\alpha, \beta$. The group $\Gamma$
  is hyperbolic relative to the subgroups $\langle\alpha\rangle$,
  $\langle \beta \rangle$, and the inclusion
  $\Gamma \hookrightarrow \SL(d + 2, \R)$ is EGF with respect to
  $P_{1,d+1}$ (the stabilizer of a line in a hyperplane in
  $\R^{d+2}$).

  Here, there are peripherally stable deformations of $\Gamma$ which
  change the Jordan block decomposition of $A$. For instance, consider
  a continuous path $A_t:[0, 1] \to \SL(d+2, \R)$ given by
  $A_t = \begin{psmallmatrix}B_t\\& 1 \\ & & 1\end{psmallmatrix}$,
  where $B_0 = B$ and $B_t$ is a \emph{diagonalizable} matrix in
  $\SL(d, \R)$. For small values of $t$, the group $\Gamma_t$
  generated by $\alpha_t = A_t^k$ and $\beta$ is still discrete and
  freely generated by $\alpha_t$ and $\beta$---since the
  ``attracting'' fixed points of $A_t$ deform continuously with $t$,
  the same exact ping-pong setup works for all small $t \ge 0$. And
  indeed the path in $\Hom(\Gamma, \SL(d+2, \R))$ determined by the
  path $A_t$ is a peripherally stable subspace.

  On the other hand, consider the path
  $A_t' = \begin{psmallmatrix}B\\& e^t\\ & &
    e^{-t}\end{psmallmatrix}$, and let $\alpha_t' = A_t'^k$. In this
  case the corresponding subspace of $\Hom(\Gamma, \SL(d+2, \R))$ is
  \emph{not} peripherally stable: while the group generated by
  $\alpha_t'$ is still discrete, the attracting fixed points of $A_t'$
  do \emph{not} deform continuously in $t$. So, there is no way to use
  the ping-pong setup for $\Gamma$ to ensure that
  $\Gamma_t' = \langle \alpha_t', \beta\rangle$ is a discrete group.
\end{example}

\begin{example}
  Here is a somewhat more interesting example of a
  \emph{non}-peripherally stable deformation. Let $M$ be a
  finite-volume noncompact hyperbolic 3-manifold, with holonomy
  representation $\rho:\pi_1M \to \PSL(2, \C)$ (so there is an
  identification $M = \H^3 / \rho(\pi_1M)$). Then $\pi_1M$ is
  hyperbolic relative to the collection $\mc{C}$ of conjugates of cusp
  groups (each of which is isomorphic to $\Z^2$), and the
  representation $\rho$ is geometrically finite (in particular, EGF).

  In this case, for \emph{any} sufficiently small nontrivial
  deformation $\rho'$ of $\rho$ in the character variety
  $\Hom(\pi_1M, \PSL(2, \C)) / \PSL(2, \C)$, the restriction of
  $\rho'$ to some cusp group $C \in \mc{C}$ either fails to be
  discrete or has infinite kernel. So $\Hom(\pi_1M, \PSL(2, \C))$ is
  \emph{not} peripherally stable, because any sufficiently small
  deformation of $\rho$ inside of a peripherally stable subspace must
  have discrete image and finite kernel on each $C \in \mc{C}$. This
  is true despite the fact that arbitrarily small deformations of
  $\rho$ are associated to complete hyperbolic structures on Dehn
  fillings of $M$.
\end{example}

The main ingredient in the proof of \Cref{thm:cusp_stable_stability}
is the relative quasigeodesic automaton $\mc{G}$ and the associated
$\mc{G}$-compatible system of open sets $\{U_v\}$ we constructed in
\Cref{prop:convergence_dynamics_implies_automaton}. The following
proposition is immediate from the definition of peripheral stability:

\begin{prop}
  Let $\rho:\Gamma \to G$ be an EGF representation with boundary
  extension $\phi$, and let $\mc{W} \subset \Hom(\Gamma, G)$ be a subspace
  which is peripherally stable at $(\rho, \phi)$.

  If $\mc{G}$ is a relative quasigeodesic automaton for $\Gamma$, and
  $\{U_v\}$ is a $\mc{G}$-compatible system of open subsets of
  $\flags$ for $\rho(\Gamma)$, then there is an open subset
  $\mc{W}' \subset \mc{W}$ containing $\rho$ such that for every
  $\rho' \in \mc{W}'$, $\{U_v\}$ is also a $\mc{G}$-compatible system
  of open sets for $\rho'(\Gamma)$.
\end{prop}

\Cref{thm:cusp_stable_stability} then follows from a kind of converse
to \Cref{prop:convergence_dynamics_implies_automaton}: we will show
that we can reconstruct a map extending the convergence dynamics of
$\Gamma$ from the $\mc{G}$-compatible system $\{U_v\}$.

\subsection{An equivariant map on conical limit points}

For the rest of this section, we let $\rho:\Gamma \to G$ be a
representation which is EGF with respect to a symmetric parabolic
subgroup $P \subset G$. We let $\phi:\Lambda \to \bgamh$ be a boundary
extension for $\rho$, and assume that $\mc{W} \subset \Hom(\Gamma, G)$
is peripherally stable at $(\rho, \phi)$. We also let $Z$ be a compact
subset of $\bgamh$, and let $V \subset \flags$ be an open subset
containing $\phi^{-1}(Z)$. We again fix a finite subset
$\Pi \subset \parbdry(\Gamma, \mc{H})$, containing one point from
every $\Gamma$-orbit in $\parbdry(\Gamma, \mc{H})$.

Using \Cref{prop:convergence_dynamics_implies_automaton}, we can find
a relative quasigeodesic automaton $\mc{G}$ and $\mc{G}$-compatible
system $\{U_v\}$ of open subsets of $\flags$ for $\rho(\Gamma)$. Using
\Cref{prop:compact_bowditch_nbhd}, we can ensure that for any
$z \in Z$, there is a $\mc{G}$-path $\{\alpha_n\}$ limiting to $z$
(with vertex path $\{v_n\}$) so that $U_{v_1}$ is contained in $V$.

Antipodality of the map $\phi$ implies that for each $z \in \bgamh$,
each fiber $\phi^{-1}(z)$ is a closed subset of some affine chart in
$\flags$. So, we can also assume that $U_v$ is a proper domain for
each vertex $v$ of $\mc{G}$. In fact, by way of the following lemma,
we can assume even more:

\begin{lem}
  \label{lem:compatible_system_transverse}
  Let $\rho$ be an EGF representation with boundary map
  $\phi:\Lambda \to \bgamh$.

  For any $\delta > 0$, we can find a relative quasigeodesic automaton
  $\mc{G}$ with $\mc{G}$-compatible system $\{U_v\}$ of open sets in
  $\flags$ as in \Cref{prop:convergence_dynamics_implies_automaton},
  so that for any $x, y \in \bgamh$ with $d(x,y) > \delta$, if
  $\phi^{-1}(x) \subset U_v$ and $\phi^{-1}(y) \subset U_w$, then
  $\overline{U_v}$ and $\overline{U_w}$ are opposite.
\end{lem}
\begin{proof}
  We choose $\eps > 0$ so that if $d(v, w) > \delta / 2$ for
  $v, w \in \bgamh$, then the closed $\eps$-neighborhoods of
  \[
    \phi^{-1}(v), \qquad \phi^{-1}(w)
  \]
  are opposite. This is possible for a fixed pair
  $v, w \in \bgamh$ since antipodality is an open condition, and
  $\phi^{-1}(v)$, $\phi^{-1}(w)$ are opposite compact sets. Then we
  can pick a uniform $\eps$ for all pairs since the the subset
  $\{(u, v) \in (\bgamh)^2: d(u,v) > \delta/2\}$ is compact.

  Consider $\mc{G}$-compatible systems of open subsets $\{U_v\}$ and
  $\{W_v\}$ for the action of $\Gamma$ on $\flags$ and
  $\bgamh$, coming from
  \Cref{prop:convergence_dynamics_implies_automaton}. We can ensure
  that for each vertex $a$, the diameter of $W_a$ is at most
  $\delta / 4$, and $U_a \subset N(\phi^{-1}(w), \eps)$ for some
  $w \in W_a$.

  If $x, y \in \bgamh$ satisfy $d(x,y) > \delta$, and
  $x \in W_a$, $y \in W_b$, we have
  \[
    d(v, w) > \delta / 2
  \]
  for all $v \in W_a$, $w \in W_b$.

  Then, if $\phi^{-1}(x) \subset U_a$ and $\phi^{-1}(y) \subset U_b$,
  we have
  \[
    U_a \subset N(\phi^{-1}(v), \eps), \qquad U_b \subset
    N(\phi^{-1}(w), \eps)
  \]
  for $v \in W_a$, $w \in W_b$ with $d(v, w) > \delta / 2$. By our
  choice of $\eps$, the closures of $N(\phi^{-1}(w), \eps)$ and
  $N(\phi^{-1}(v), \eps)$ are opposite.
\end{proof}

Using cocompactness of the action of $\Gamma$ on the space of distinct
pairs in $\bgamh$, we know that there exists some fixed
$\delta > 0$ such that for any distinct
$z_1, z_2 \in \bgamh$, we can find some
$\gamma \in \Gamma$ such that $d(\gamma z_1, \gamma z_2) >
\delta$. Then, in light of \Cref{lem:compatible_system_transverse}, we
can make the following assumption:
\begin{assumption}
  \label{ass:opposite_open_sets}
  For any $z_1, z_2 \in \bgamh$ satisfying $d(z_1, z_2) > \delta$, if
  $\phi^{-1}(z_1) \subset U_v$ and $\phi^{-1}(z_2) \subset U_w$ for
  $v,w$ vertices of $\mc{G}$, then $\overline{U_v}$ and
  $\overline{U_w}$ are opposite.
\end{assumption}

With our relative quasigeodesic automaton $\mc{G}$ and compatible
system of open sets $\{U_v\}$ fixed, we now choose an open subset
$\mc{W}' \subset \mc{W}$ so that for any $\rho' \in \mc{W}'$,
$\{U_v\}$ is also a $\mc{G}$-compatible system for the action of
$\rho'(\Gamma)$ on $\flags$. Our main goal for the rest of this
section is to show that any $\rho' \in \mc{W}'$ is an EGF
representation. So, we fix some $\rho' \in \mc{W}'$.

Let $\mr{Path}(\mc{G})$ denote the set of infinite $\mc{G}$-paths.
\Cref{prop:compatible_system_in_flags_contracting} implies that every
path in $\mr{Path}(\mc{G})$ is contracting for the $\rho'$-action, so
we have a map
\[
  \psi_{\rho'}:\mr{Path}(\mc{G}) \to \flags,
\]
where the path $\{\alpha_n\}$ maps to the unique element of
\[
  \bigcap_{n=1}^\infty \rho'(\alpha_1) \cdots \rho'(\alpha_n)
  U_{v_{n+1}}.
\]

\begin{lem}
  \label{lem:equivariant_conical_map}
  The map $\psi_{\rho'}:\mr{Path}(\mc{G}) \to \flags$ induces an
  equivariant map
  \[
    \psi_{\rho'}:\conbdry \Gamma \to \flags.
  \]
\end{lem}
\begin{proof}
  We first need to see that $\psi_{\rho'}$ is well-defined, i.e. that
  if $z$ is a conical limit point and $\{\alpha_n\}$, $\{\beta_n\}$
  are $\mc{G}$-paths limiting to $z$, then
  $\psi_{\rho'}(\{\alpha_n\}) = \psi_{\rho'}(\{\beta_n\})$.

  Let
  \[
    \gamma_n = \alpha_1 \cdots \alpha_n, \qquad \eta_m = \beta_1
    \cdots \beta_m.
  \]
  We can use \Cref{prop:bowditch_graphs_geodesic} to see that
  $\gamma_n$ and $\eta_m$ lie within bounded Hausdorff distance of a
  geodesic in $\conedgraph$ limiting to $z$, so there is a fixed $D$
  so that for infinitely many pairs $m,n$,
  \[
    d(\gamma_n, \eta_m) < D
  \]
  in the Cayley graph of
  $\Gamma$. \Cref{lem:graph_contracting_implies_contracting} implies
  that $\rho'(\gamma_n)$ and $\rho'(\eta_n)$ are both $P$-divergent
  sequences and each have a unique $P$-limit point in $\flags$, given
  by $\psi_{\rho'}(\{\alpha_n\})$, $\psi_{\rho'}(\{\beta_m\})$,
  respectively. Then, Lemma 4.23 in \cite{klp2017anosov} implies that
  because $\rho'(\gamma_n) = \rho'(\eta_n)g_n$ for a bounded sequence
  $g_n \in G$, the $P$-limit points of $\rho'(\gamma_n)$ and
  $\rho'(\eta_n)$ must agree and therefore
  $\psi_{\rho'}(\{\alpha_n\}) = \psi_{\rho'}(\{\beta_m\})$.

  Next we observe that $\psi_{\rho'}$ is equivariant. Fix a finite
  generating set $S$ for $\Gamma$. It suffices to show that
  $\psi_{\rho'}(s \cdot z) = \rho'(s) \cdot \psi_{\rho'}(z)$ for all
  $s \in S$.

  Let $\{\alpha_n\}$ be a $\mc{G}$-path limiting to some
  $z \in \conbdry\Gamma$, and consider the sequence
  \[
    \gamma_n' = s\alpha_1 \cdots \alpha_n.
  \]
  
  Again, \Cref{prop:bowditch_graphs_geodesic} implies that $\gamma_n'$
  lies bounded Hausdorff distance from a geodesic in $\conedgraph$,
  which must limit to $s \cdot z$. So if we fix a $\mc{G}$-path
  $\beta_n$ limiting to $s \cdot z$, the same argument as above shows
  that
  $\psi_{\rho'}(\{\beta_n\}) = \rho'(s) \cdot
  \psi_{\rho'}(\{\alpha_n\})$.
\end{proof}

It will turn out that $\psi_{\rho'}$ is also both continuous and
injective. However, we do not prove this directly.

\subsection{Extending $\psi_{\rho'}$ to parabolic points}

We want to extend the map $\psi_{\rho'} : \conbdry\Gamma \to \flags$
to the entire Bowditch boundary $\bgamh$. To do so, we need to
view $\psi_{\rho'}$ as a map to the set of closed subsets of $\flags$.

The first step is to define $\psi_{\rho'}$ on the finite set
$\Pi \subset \parbdry\Gamma$. For any vertex $v$ in $\mc{G}$, we
consider the set
\[
  B_v = \bigcup_{(v, y) \textrm{ edge in } \mc{G}} U_y.
\]
Then, for each $p \in \Pi$, we pick a parabolic vertex $w$ so that
$p_w = p$. We define $\Lambda_p'$ to be the closure of the set of
accumulation points of sequences of the form
$\rho'(\gamma_n) \cdot x$, for $x \in B_w$ and $\gamma_n$ distinct
elements of $\Gamma_{p}$. Part (3) of
\Cref{prop:convergence_dynamics_implies_automaton} guarantees that
$B_w \subset C_p$, and $\mc{G}$-compatiblity of the system $\{U_v\}$
for the $\rho'(\Gamma)$-action on $\flags$ implies that
$\Lambda_{p}' \subset U_w$. By construction, $\Lambda_{p}'$ is
invariant under the action of $\rho(\Gamma_p)$.

Next, given a parabolic point $q \in \parbdry \Gamma$, we write
$q = g \cdot p$ for $p \in \Pi$, and then define
\[
  \psi_{\rho'}(q) := \rho'(g) \Lambda_p'.
\]
Since $\Lambda_p'$ is $\Gamma_p$-invariant and $\Gamma_p$ is exactly
the stabilizer of $p$, this does not depend on the choice of coset
representative in $g\Gamma_p$. Moreover $\psi_{\rho'}$ is still
$\rho'$-equivariant.

In addition, if $v$ is any parabolic vertex with parabolic point
$p_v = g \cdot p$ for $p \in \Pi$, part (2) of
\Cref{prop:convergence_dynamics_implies_automaton} ensures that
$B_v = B_w$ for any parabolic vertex $w$ with $p_w = p$. So,
$\rho'(g) \cdot \Lambda_p'$ is exactly the closure of the set of
accumulation points of the form $\rho'(g\gamma_n)\cdot x$ for
sequences $\gamma_n \in \Gamma_p$ and $x \in B_v$. Then
$\mc{G}$-compatibility implies that
$\psi_{\rho'}(p_v) = \rho(g)\Lambda_p'$ is a subset of $U_v$.

\begin{remark}
  There is a natural topology on the space of closed subsets of
  $\flags$, induced by the Hausdorff distance arising from some (any)
  choice of metric on $\flags$. We emphasize that the map
  $\psi_{\rho'}$ is \emph{not} necessarily continuous with respect to
  this topology.
\end{remark}

Ultimately we want to use $\psi_{\rho'}$ to define a map extending the
convergence dynamics of $\Gamma$, so we will need to also define the
sets $C_z'$ for each $z \in \bgamh$. For now, we only define
$C_p'$ for $p \in \Pi$: this will be the set
\[
  \bigcup_{ \gamma \in \Gamma_p} \rho'(\gamma) B_w.
\]
We can immediately observe:
\begin{prop}
  \label{prop:convergence_dynamics_deformed_parabolics}
  $C_p'$ is $\rho'(\Gamma_p)$-invariant. Moreover, for any infinite
  sequence $\gamma_n \in \Gamma_p$, any compact $K \subset C_p'$, and
  any open $U \subset \flags$ containing $\Lambda_p'$, for
  sufficiently large $n$, $\rho'(\gamma_n) \cdot K$ lies in $U$.
\end{prop}
\begin{proof}
  $\Gamma_p$-invariance follows directly from the definition.

  Fix a compact $K \subset C_p'$ and an open $U \subset \flags$
  containing $\Lambda_p'$. $K$ is contained in finitely many sets
  $\rho'(\gamma) B_w$ for $\gamma \in \Gamma_p$, so any accumulation
  point of $\rho'(\gamma_n) x$ for $x \in K$ and
  $\gamma_n \in \Gamma_p$ lies in $\Lambda_p'$. In particular, for
  sufficiently large $n$, $\gamma_n x$ lies in $U$, and since $K$ is
  compact we can pick $n$ large enough so that $\gamma_n x \in U$ for
  all $x \in K$.
\end{proof}

We next want to use $\psi_{\rho'}$ to define an antipodal extension
from a subset of $\flags$ to $\bgamh$.

\begin{lem}
  \label{lem:initial_vertex_neighborhood}
  For any $z \in \bgamh$, if $\{\alpha_n\}$ is a $\mc{G}$-path
  limiting to $z$ with corresponding vertex path $\{v_n\}$, then
  $\phi^{-1}(z)$ and $\psi_{\rho'}(z)$ are both subsets of $U_{v_1}$.
\end{lem}
\begin{proof}
  If $z$ is a conical limit point, then this follows immediately from
  \Cref{prop:bowditch_graphs_geodesic} and the definition of
  $\psi_{\rho'}$. On the other hand, if $z$ is a parabolic point, then
  $z = \alpha_1 \cdots \alpha_N p_v$, where $v$ is a parabolic vertex
  at the end of the vertex path $\{v_n\}$. By part (3) of
  \Cref{prop:convergence_dynamics_implies_automaton}, we have
  $p_v \in W_v$ and thus $\phi^{-1}(p_v) \subset U_v$. By
  $\rho$-equivariance of $\phi$ we have
  \[
    \phi^{-1}(z) = \rho(\alpha_1 \cdots \alpha_N) \phi^{-1}(p_v),
  \]
  so by $\mc{G}$-compatibility we have $\phi^{-1}(z) \subset
  U_{v_1}$. On the other hand, we have constructed $\psi_{\rho'}$ so
  that $\psi_{\rho'}(p_v) \subset U_v$, so $\rho'$-equivariance of
  $\psi_{\rho'}$ and $\mc{G}$-compatibility also show that
  $\psi_{\rho'}(z) \subset U_{v_1}$.
\end{proof}

\begin{lem}
  \label{lem:boundary_map_well_defined}
  For any two distinct points $z_1, z_2$ in $\bgamh$, the sets
  \[
    \psi_{\rho'}(z_1), \qquad \psi_{\rho'}(z_2)
  \]
  are opposite (in particular disjoint).
\end{lem}
\begin{proof}
  We know that for any distinct $z_1, z_2 > 0$, we can find
  $\gamma \in \Gamma$ so that $d(\gamma z_1, \gamma z_2) >
  \delta$. So, since $\psi_{\rho'}$ is $\rho'$-equivariant, we just
  need to show that if $z_1, z_2 \in \bgamh$ satisfy
  $d(z_1, z_2) > \delta$, then $\psi_{\rho'}(z_1)$ is opposite to
  $\psi_{\rho'}(z_2)$.

  Let $\{\alpha_n\}$, $\{\beta_n\}$ be $\mc{G}$-paths respectively
  limiting to points $z_1, z_2 \in \bgamh$ with
  $d(z_1, z_2) > \delta$, with corresponding vertex paths $\{v_n\}$
  and $\{w_n\}$. By \Cref{lem:initial_vertex_neighborhood}, we must
  have $\phi^{-1}(z_1) \subset U_{v_1}$ and
  $\phi^{-1}(z_2) \subset U_{w_2}$, so under
  \Cref{ass:opposite_open_sets}, we know that $U_{v_1}$ and $U_{w_1}$
  are opposite. But then we are done since
  \Cref{lem:initial_vertex_neighborhood} also implies that
  $\psi_{\rho'}(z_1) \subset U_{v_1}$ and
  $\psi_{\rho'}(z_2) \subset U_{w_1}$.
\end{proof}

\subsection{The boundary set of the deformed representation}

We define our candidate boundary set $\Lambda' \subset \flags$ by
\[
  \Lambda' = \bigcup_{z \in \bgamh} \psi_{\rho'}(z).
\]
We then have an equivariant map
\[
  \phi':\Lambda' \to \bgamh,
\]
where $\phi'(x) = z$ if $x \in
\psi_{\rho'}(z)$. \Cref{lem:boundary_map_well_defined} implies that
$\phi'$ is well-defined and antipodal. It is necessarily both
surjective and $\rho'$-equivariant, and its fibers are either
singletons or translates of the sets $\Lambda_p'$ for $p \in \Pi$.

\begin{lem}
  For any $z \in Z$, we have $\phi'^{-1}(z) \subset V$.
\end{lem}
\begin{proof}
  Recall that we used \Cref{prop:compact_bowditch_nbhd} to construct
  our automaton so that for any $z \in Z$, there is a $\mc{G}$-path
  limiting to $z$ with vertex path $\{v_n\}$ such that
  $U_{v_1} \subset V$. Then \Cref{lem:initial_vertex_neighborhood}
  implies $\phi'^{-1}(z) \subset V$.
\end{proof}

\begin{lem}
  \label{lem:deformed_map_proper}
  $\Lambda'$ is compact.
\end{lem}
\begin{proof}
  Fix a sequence $y_n \in \Lambda'$, and let $x_n = \phi'(y_n)$. Since
  $\bgamh$ is compact, up to subsequence $x_n \to x$. We want to
  see that a subsequence of $y_n$ converges to some $y \in
  \Lambda'$. We consider two possibilities:
  \begin{aside}{Case 1: $x$ is a parabolic point}
    We can write $x = g \cdot p$, where $p \in \Pi$. Let $w$ be a
    parabolic vertex with $p_w = p$.

    Choose an edge $(w, v)$ in $\mc{G}$. Part (3) of
    \Cref{prop:convergence_dynamics_implies_automaton} implies that we
    can find a compact subset $K \subset W_v$ such that
    $\Gamma_p \cdot K = \bgamh - \{p\}$. So, we can find
    $\gamma_n \in \Gamma_p$ so that
    \[
      z_n = \gamma_n^{-1}g^{-1} x_n \in K.
    \]

    Then, since $\phi^{-1}(W_v) \subset U_v$, by definition $C_p'$
    contains $\phi'^{-1}(K)$, and in particular $C_p'$ contains
    $\phi'^{-1}(z_n)$ for infinitely many $z_n$.
    
    Then using \Cref{prop:convergence_dynamics_deformed_parabolics},
    we know that up to subsequence,
    \[
      \rho'(\gamma_n) \phi'^{-1}(z_n) = \rho'(\gamma_n)
      \phi'^{-1}(\gamma_n^{-1} g^{-1}x_n)
    \]
    converges to a compact subset of $\Lambda_p'$, which means that
    \[
      y_n \in \rho'(g)\rho'(\gamma_n) \phi'^{-1}(\gamma_n^{-1} g^{-1} x_n)
    \]
    subconverges to a point in $\rho'(g) \Lambda_p'$.
  \end{aside}
  \begin{aside}{Case 2: $x$ is a conical limit point}
    We want to show that any sequence in $\phi'^{-1}(x_n)$ limits to
    $\phi'^{-1}(x)$, so fix any $\eps > 0$. Using
    \Cref{cor:exponential_path_shrinking}, we can choose $N$ so that
    if $\{\alpha_m\}$ is any $\mc{G}$-path limiting to $x$, with
    corresponding vertex path $\{v_m\}$, then the diameter of
    \[
      \rho'(\alpha_1 \cdots \alpha_N) U_{v_{N+1}}
    \]
    is less than $\eps$ with respect to a metric on $U_{v_1}$. We fix
    such a $\mc{G}$-path $\{\alpha_m\}$. Then, we use
    \Cref{lem:close_points_share_prefixes} to see that for
    sufficiently large $n$, there is a $\mc{G}$-path $\{\beta^n_m\}$
    limiting to $x_n$ with $\beta_i = \alpha_i$ for $i \le N$. Thus
    the Hausdorff distance (with respect to $C_{U_{v_1}}$) between
    $\phi'^{-1}(x_n)$ and $\phi'^{-1}(x)$ is at most $\eps$. Since
    $\phi'^{-1}(x_n)$ and $\phi'^{-1}(x)$ both lie in the compact set
    $\rho'(\alpha_1)\overline{U_{v_2}} \subset U_{v_1}$, this proves
    the claim.
  \end{aside}
\end{proof}

\begin{lem}
  $\phi'$ is continuous and proper.
\end{lem}
\begin{proof}
  Since $\Lambda'$ is compact, we just need to show continuity. Fix
  $y \in \Lambda'$ and a sequence $y_n \in \Lambda'$ approaching
  $y$. We want to show that $\phi'(y_n)$ approaches $\phi'(y) = x$.

  Suppose otherwise. Since $\bgamh$ is compact, up to a
  subsequence $z_n = \phi'(y_n)$ approaches $z \ne x$. Using the
  equivariance of $\phi'$, and cocompactness of the $\Gamma$-action on
  distinct pairs in $\bgamh$, we may assume that
  $d(x, z) > \delta$. For sufficiently large $n$, we have
  $d(x, z_n) > \delta$ as well. Then, as in the proof of
  \Cref{lem:boundary_map_well_defined}, by
  \Cref{ass:opposite_open_sets} we know that for any vertices $v,w$ in
  $\mc{G}$ such that $U_v$ contains $\psi_{\rho'}(x)$ and $U_w$
  contains $\psi_{\rho'}(z_n)$, the intersection
  $\overline{U_v} \cap \overline{U_w}$ is empty.

  But by definition of $\phi'$, we have
  \[
    y \in \psi_{\rho'}(x) \subset U_v, \qquad y_n \in
    \psi_{\rho'}(z_n) \subset U_w
  \]
  for vertices $v, w$ in $\mc{G}$. This contradicts the fact that
  $y_n \to y$.
\end{proof}

\subsection{Dynamics on the deformation}

To complete the proof of \Cref{thm:cusp_stable_stability}, we just
need to show:

\begin{prop}
  \label{prop:deformed_map_weakly_dynamics_preserving}
  $\phi'$ extends the convergence dynamics of $\Gamma$.
\end{prop}
\begin{proof}
  We will apply \Cref{prop:conical_peripheral_implies_egf}, and show
  separately that the required dynamics hold for conical limit
  sequences and sequences in peripheral subgroups.
  
  For each conical limit point $z$, we define
  $C_z' = \Opp(\phi'^{-1}(z))$. Antipodality of $\phi'$ implies that
  $C_z'$ contains $\Lambda' - \phi'^{-1}(z)$ for all
  $z \in \conbdry\Gamma$.

  First suppose that $\gamma_n$ is a sequence limiting conically to a
  point $z \in \bgamh$, and that $\gamma_n^{-1}$ limits conically to
  some $z_- \in \bgamh$.

  Up to subsequence, $\gamma_n$ lies a bounded distance away from the
  image of a $\mc{G}$-path $\{\alpha_n\}$ limiting to
  $z$. \Cref{lem:graph_contracting_implies_contracting} implies that
  $\rho'(\gamma_n)$ is $P$-divergent, with unique $P$-limit
  $\phi'^{-1}(z)$. Similarly, $\rho'(\gamma_n^{-1})$ is $P$-divergent,
  with unique $P$-limit $\phi'^{-1}(z_-)$. Then
  \Cref{lem:klp_contraction_symmetry} implies that $\rho'(\gamma_n)$
  converges uniformly to $\phi'^{-1}(z)$ on compacts in $C_{z_-}'$.

  For each parabolic point $q$, we write $q = g \cdot p$ for
  $p \in \Pi$, and then take $C_q' = \rho'(g) \cdot
  C_p'$. \Cref{prop:convergence_dynamics_deformed_parabolics} says
  that for any $p \in \Pi$, any compact $K \subset C_p'$, and any open
  $U \subset \flags$ containing $\Lambda_p'$, if $\gamma_n$ is an
  infinite sequence in $\Gamma_p$, then
  $\rho'(\gamma_n) \cdot K \subset U$ for sufficiently large
  $n$. Then, since $\Gamma_q = g\Gamma_pg^{-1}$, the same is true for
  any parabolic point $q$.

  So, we just need to check that for each $p \in \Pi$, $C_p'$ contains
  $\Lambda' - \Lambda_p'$. As in the proof of
  \Cref{lem:deformed_map_proper}, we can find a compact subset
  $K \subset \bgamh - \{p\}$ so that
  $\Gamma_pK = \bgamh - \{p\}$ and $C_p'$ contains
  $\phi'^{-1}(K)$. But then since $C_p'$ is
  $\rho'(\Gamma_p)$-invariant (by
  \Cref{prop:convergence_dynamics_deformed_parabolics}), we have
  \[
    \rho'(\Gamma_p) \cdot \phi'^{-1}(K) = \phi'^{-1}(\bgamh -
    \{p\}) \subset C_p'.
  \]
\end{proof}

\begin{remark}
  \label{rem:conical_pts_singletons}
  The definition of the set $\Lambda'$ and the map $\phi'$ immediately
  imply that the \emph{fibers} of the deformed boundary extension
  $\phi':\Lambda' \to \bgamh$ satisfy the conclusions of
  \Cref{prop:conical_pts_singletons}: the fiber over each conical
  limit point is a singleton, and the fiber over each parabolic point
  $p$ is the closure of the accumulation sets of $\Gamma_p$-orbits in
  $C_p'$. So, we obtain \Cref{prop:conical_pts_singletons} by taking
  $\mc{W}$ to be the singleton $\{\rho\}$, and following the proof of
  \Cref{thm:cusp_stable_stability} (using $C_p$ for $C_p'$
  throughout).
\end{remark}


\subsection{Symmetric powers}

Let $V$ be a finite-dimensional real vector space. We let $\tau_m$
denote the symmetric representation
\[
  \SL(V) \to \SL(\Sym^m(V)).
\]
Throughout this section, we will view $\Sym^m(V)$ as a quotient of the
space of homogeneous degree-$m$ polynomials in elements of $V$. We
will always leave this quotient implicit. That is, if
$v_1, \ldots, v_k \in V$, and $r_1, \ldots, r_k \in \N \cup \{0\}$
with $\sum r_i = m$, we will view the monomial
$v_1^{r_1}\cdots v_k^{r_k}$ as an element of $\Sym^m(V)$.

There is a $\tau_m$-equivariant embedding
\[
  \iota:\P(V) \to \P(\Sym^m(V))
\]
given by $[v] \mapsto [v^m]$. There is also a corresponding dual
embedding
\[
  \iota^*:\P(V^*) \to \P(\Sym^m(V)^*),
\]
using the canonical identification $\Sym^m(V^*) \simeq \Sym^m(V)^*$.
We observe that $v \in \P(V)$ and $w \in \P(V^*)$ are transverse if
and only if their respective images under $\iota$ and $\iota^*$ are
also transverse. This means that the maps $\iota$ and $\iota^*$ also
give rise to a $\tau_m$-equivariant map
\[
  \hat{\iota}:\projflags(V) \to \mc{F}_{1,d'}(\Sym^m(V))
\]
given by $\hat{\iota}(v, w) = (\iota(v), \iota^*(w))$. Here, $d + 1$
is the dimension of $V$ and $d' + 1$ is the dimension of
$\Sym^m(V)$. In this section we will mostly use will use
$\symflags(\Sym^mV)$ to denote $\mc{F}_{1,d'}(\Sym^mV)$.

\subsubsection{Dynamics in symmetric powers}

The dynamics of divergent sequences in $\SL(V)$ on $\P(V)$ and
$\P(V^*)$ are respected in $\P(\Sym^mV)$, in the following sense:
\begin{prop}
  \label{prop:symmetric_power_dynamics}
  Let $\{g_n\}$ be an infinite sequence in $\SL(V)$ such that for some
  $w \in \P(V^*)$, $x \in \P(V)$, we have $g_n|_{\Opp(w)} \to x$
  uniformly on compacts.
  
  Then
  \[
    \tau_m(g_n)|_{\Opp(\iota^*(w))} \to \iota(x)
  \]
  uniformly on compacts.
\end{prop}

\subsection{Symmetric powers of relatively hyperbolic convex cocompact
  groups}

Suppose that $\overline{\rho}:\Gamma \to \PGL(V)$ is a convex
cocompact representation, so that there is a properly convex domain
$\Omega \subset \P(V)$ with $\overline{\rho}(\Gamma)$ acting convex
cocompactly on $\Omega$. As mentioned in the introduction to this
paper, we can always replace $\Gamma$ with a finite-index subgroup so
that $\overline{\rho}$ lifts to a representation
$\rho:\Gamma \to \SL(V)$ with finite kernel and discrete image. In
this situation, we also say that $\rho:\Gamma \to \SL(V)$ is
\emph{convex cocompact in $\P(V)$}.

Let $(\Gamma, \mc{H})$ be a relatively hyperbolic pair such that each
group in $\mc{H}$ is virtually abelian, and let
$\rho:\Gamma \to \SL(V)$ be a convex cocompact
representation. Representations of this form have been studied
extensively by Islam-Zimmer \cite{iz2019convex}, \cite{iz2019flat},
who proved a number of strong structural results. In particular,
Islam-Zimmer showed that in this situation, for each $H \in \mc{H}$,
the image $\rho(H)$ acts cocompactly on a properly embedded
$k$-simplex $\Delta_H \subset \Omega$, with $k = \mathrm{rank}(H)$.(A
simplex $\Delta \subset \Omega$ is \emph{properly embedded} if
$\dee \Delta \subset \dee \Omega$.) In fact, the action of $\rho(H)$
on $\Omega$ is convex cocompact, and the full orbital limit set of
$\rho(H)$ in $\Omega$ is $\dee \Delta_H$.

Conversely, every properly embedded maximal $k$-simplex in the convex
hull of $\Lambda_\Omega(\Gamma)$ always has a cocompact action by some
$H \in \mc{H}$ with rank $k$.

We let
\[
  \rho^m: \Gamma \to \SL(\Sym^mV)
\]
denote the composition $\tau_m \circ \rho$. We have two main goals in
this section. The first is to prove that the representation $\rho^m$
is always EGF (with respect to the parabolic stabilizing a line in a
hyperplane). This result does \emph{not} follow directly from the fact
that convex cocompact representations in $\P(V)$ are EGF
(\Cref{thm:convex_cocompact_boundary_dynamics}), because we do not
know that the representations $\rho^m$ are convex cocompact in
$\P(\Sym^mV)$. In fact, Jeff Danciger and Fanny Kassel have indicated
in personal communication to the author that $\rho^m$ should
\emph{never} be convex cocompact in $\P(\Sym^mV)$ unless the
collection $\mc{H}$ is empty: while $\rho^m(\Gamma)$ does preserve a
properly convex domain in $\P(\Sym^mV)$, the convex hull of the full
orbital limit set in any such domain seems ``too big'' for
$\rho^m(\Gamma)$ to act cocompactly.

Our second goal is to show that the \emph{entire} space
$\Hom(\Gamma, \SL(\Sym^mV))$ is peripherally stable about $\rho^m$,
meaning (by \Cref{thm:cusp_stable_stability}) an open neighborhood of
$\rho^m$ consists of EGF representations. In particular this shows
that small perturbations of $\rho^m$ still have finite kernel and
discrete image, giving new examples of discrete subgroups of
higher-rank Lie groups which are stable (as discrete groups).

\subsubsection{Proof strategy}

To show that $\rho^m$ is EGF, we will give an explicit description of
the boundary set $\hat{\Lambda}_m \subset \symflags(\Sym^mV)$. The
naive choice is to just take $\hat{\Lambda}_m$ to be
$\hat{\iota}(\hat{\Lambda}_\Omega(\Gamma))$, where
$\hat{\Lambda}_\Omega(\Gamma)$ is the flag-valued full orbital limit
set giving the EGF boundary set for $\rho:\Gamma \to \SL(V)$ (see
\Cref{sec:convex_projective_egf}). While there is a equivariant
surjective map from this set to $\bgamh$, it turns out that we will
have to \emph{enlarge} it in order to ensure that the relevant
dynamics hold.

The idea is the following: for each parabolic point $p \in \bgamh$,
with stabilizer $H$, we take the fiber over $p$ in $\hat{\Lambda}_m$
to be the space of flags in the boundary of a simplex
$S_H \subset \P(\Sym^mV)$, constructed using the simplex
$\Delta_H \subset \Omega$ on which $H = \Stab_\Gamma(p)$ acts
cocompactly. The simplex $S_H$ is chosen so that if $\gamma_n$ is a
sequence in $\Gamma$ converging to $p$, then a face of $S_H$ spans a
minimal attracting subspace of $\rho^m(\gamma_n)$. We also want to
ensure that the simplex $S_H$ is \emph{stable}, i.e. if $\rho^m_t$ is
a small deformation of $\rho^m$ in $\Hom(\Gamma, \SL(\Sym^mV))$, then
$\rho_t^m(H)$ preserves a simplex $S_H^t$ close to $S_H$. We verify
that $S_H$ has these properties by analyzing the relationship between
the weights of $\rho$ and $\rho^m$ on the virtually abelian group $H$.

The other main steps in the proof involve checking that the boundary
set $\hat{\Lambda}_m$ we construct is actually a compact space
surjecting continuously onto $\bgamh$, and constructing the open sets
$\hat{C}_p$ also required by the definition of an EGF
representation. For the latter, we make heavy use of the fact that the
\emph{dual} action of $\rho(\Gamma)$ on $\P(V^*)$ is also projectively
convex cocompact, which allows us to construct a stable \emph{dual}
simplex $S_H^*$ for each $H \in \mc{H}$. The vertices of $S_H^*$ are
thought of as hyperplanes in $\P(\Sym^m(V))$, cutting out a region
$C_p$ of $\P(\Sym^m(V))$ on which $\rho^m(H)$ attracts points towards
$S_H$.

\subsubsection{Example: symmetric squares of convex projective
  3-manifold groups}

We illustrate the general idea of our approach with a specific
example. Let $\Omega \subset \RP^3$ be a properly convex domain, and
let $\Gamma \subseteq \Aut(\Omega)$ be a discrete group acting
cocompactly on $\Omega$. In \cite{benoist2006convexes}, Benoist
produced examples of such groups which are hyperbolic relative to a
nonempty collection $\mc{H}$ of virtually abelian subgroups of rank
2. Further examples were constructed by Ballas-Danciger-Lee in
\cite{bdl2015convex}. Up to finite index, each $H \in \mc{H}$ acts
diagonalizably on $\RP^3$, preserving a projective tetrahedron
$T_H \subset \RP^3$ and acting cocompactly on a properly embedded
triangle $\Delta_H \subset \Omega$. Each edge of $\Delta_H$ is
contained in a unique supporting hyperplane of $\Omega$. The common
intersection of these hyperplanes is the fourth vertex of $T_H$.

More explicitly, up to finite index, each $H$ acts diagonally on
$\R^4$ in the basis $\{v_1, v_2, v_3, v_4\}$, where the $v_i$ are the
vertices of $T_H$, and $v_1, v_2, v_3$ are the vertices of
$\Delta_H$. We can consider the situation where (in this basis) $H$ is
the discrete group
\[
  \left\{
    \begin{psmallmatrix}
      2^a\\& 2^b\\ & & 2^c\\ & & & 1
    \end{psmallmatrix} : a,b,c \in \Z, \,a + b + c = 0 \right\}.
\]

The dual of $H$ preserves the corresponding dual basis
$\{v_1^*, v_2^*, v_3^*, v_4^*\}$, and acts cocompactly on a projective
triangle $\Delta_H^* \subset (\RP^3)^*$ with vertices
$v_1^*, v_2^*, v_3^*$. The kernels $\hyp{v_i^*}$ for $i = 1,2,3$ give
three supporting hyperplanes of $\Omega$ which cut out a region $R_H$
of projective space containing $\Omega$. In fact, $R_H$ also contains
$\dee \Omega - \dee\Delta_H$.

Now let $\rho^2:\Gamma \to \SL(\Sym^2(\R^4)) \simeq \SL(10, \R)$ be
the composition of the inclusion $\Gamma \hookrightarrow \SL(4, \R)$
with the symmetric square $\tau_2:\SL(4, \R) \to
\SL(\Sym^2(\R^4))$. In this case, the induced map
$\iota:\RP^3 \to \RP^9$ is the Veronese embedding.

For each $H \in \mc{H}$, $\rho^2(H)$ preserves a 9-simplex in $\RP^9$,
with vertices
\[
  \{v_1^2, v_2^2, v_3^2, v_4^2, v_1v_2, v_1v_3, v_1v_4, v_2v_3,
  v_2v_4, v_3v_4\}.
\]

In particular $\rho^2(H)$ also preserves the 5-simplex $S_H$ with
vertices
\[
  \{v_1^2, v_2^2, v_3^2, v_1v_2, v_2v_3, v_1v_3\}.
\]
Any divergent sequence in $\rho^2(H)$ always has an attracting
subspace spanned by a face of $S_H$, since the eigenvalues of elements
of $H$ on $v_4$ are always dominated by some eigenvalue of that
element on either $v_1, v_2$, or $v_3$. For instance, if we consider
the sequence
\[
  a_n =
  \begin{psmallmatrix}
    2^{2n}\\ & 2^{-n} \\ & & 2^{-n} \\ & & & 1
  \end{psmallmatrix},
\]
then $\rho^2(a_n)$ attracts towards the subspace spanned by
$\{v_1^2\}$. On the other hand, if $a_n$ is the sequence
\[
  \begin{psmallmatrix}
    2^n\\ & 2^n \\ & & 2^{-2n} \\ & & & 1
  \end{psmallmatrix},
\]
then $\rho^2(a_n)$ attracts towards the subspace spanned by
$\{v_1^2, v_2^2, v_1v_2\}$.

Moreover, the subspaces spanned by faces of $S_H$ are transverse to
$\iota(\dee \Omega - \dee \Delta_H)$, so large elements of $H$ attract
points in $\iota(\dee \Omega)$ that are ``far from'' $\dee S_H$
towards $\dee S_H$. In fact, this dynamical behavior extends to an
entire open subset of $\RP^9$: namely, a region cut out by the
hyperplanes corresponding to the vertices of the dual 5-simplex
$S_H^* \subset (\RP^9)^*$ with vertices
\[
  \{(v_1^*)^2, (v_2^*)^2, (v_3^*)^2, v_1^*v_2^*, v_2^*v_3^*,
  v_1^*v_3^*\}.
\]
So, the simplex $S_H$ serves as the ``parabolic point'' for the action
of the peripheral subgroup $H$ on $\RP^9$---and moreover, this
behavior is stable under perturbations of $\rho^2(H)$ in
$\Hom(\Gamma, \SL(\Sym^2V))$. This means that we can construct our
candidate boundary set for the representation $\rho^m$ by taking
\[
  \hat{\iota}(\hat{\Lambda}_\Omega(\Gamma)) \cup \bigcup_{H \in
    \mc{H}} \dee\hat{S}_H,
\]
where $\dee\hat{S}_H \subset \symflags(\Sym^2\R^4)$ is a closed subset
of the space of flags projecting to the boundary of the simplex $S_H$.

\subsection{Generalized weight spaces}

To carry out the general construction of the simplex $S_H$ identified
in the previous example, we need some description of \emph{attracting
  subspaces} for the groups $\rho^m(H) \subset \SL(\Sym^mV)$. We
obtain this description by recalling some of the properties of
\emph{weights} of representations of free abelian groups.

\begin{definition}
  Let $\rho:H \to \GL(V)$ be a representation of a free abelian group
  $H$, and let $\rho_\C:H \to \GL(V \ten \C)$ be the complexification
  of $\rho$.

  A \emph{complex weight} of $\rho$ is a homomorphism
  $\mu_\C : H \to \C$ such that the \emph{weight space}
  \[
    V_{\mu_\C} = \ker(\rho_\C(h) - \exp(\mu_\C(h))I)
  \]
  is nontrivial for some (any) $h \in H$. A \emph{generalized complex
    weight} is similarly a homomorphism $\mu_\C:H \to \C$ such that
  the \emph{generalized weight space}
  \[
    V_{\mu_\C} = \bigcup_{n=1}^{\dim V} \ker(\rho_\C(h) -
    \exp(\mu_\C(h))I)^n
  \]
  is nontrivial.

  For any generalized weight $\mu_{\C}$, the \emph{nilpotence degree}
  of $\mu_\C$ is the minimal $\ell \in \N$ such that
  $V_{\mu_\C} = \ker(\rho_\C(h) - \exp(\mu_\C(h))I)^\ell$.
\end{definition}

Given a representation $\rho:H \to \GL(V)$, the generalized complex
weight spaces of $\rho_\C$ give a $\rho_\C$-invariant decomposition of
$V \ten \C$. This in turn gives a $\rho$-invariant decomposition of
$V$, since when $\mu_\C$ is a weight which takes on complex values,
the direct sum $V_{\mu_\C} \oplus V_{\overline{\mu_\C}}$ is a
$\rho$-invariant real subspace of $V$. By a slight abuse of
terminology we refer to this as the \emph{generalized weight space
  decomposition} for the representation $\rho$. The associated
\emph{real weights} of the representation are homomorphisms
$\mu:H \to \R$ of the form $\log |\exp \mu_\C |$, where $\mu_\C$ is a
(generalized) complex weight. For the rest of the section, unless
otherwise indicated, when we refer to (generalized) \emph{weights} of
a representation into $\SL(V)$, we will mean the (generalized)
\emph{real} weights, and similarly for weight spaces.

Generalized weight spaces of $\rho$ are stable under deformations of
$\rho$. To be precise, we observe the following:
\begin{prop}
  \label{prop:weights_deform_continuously}
  Let $\rho:\Gamma \to \GL(V)$ be a representation of a free abelian
  group, and let
  \[
    V_{\mu_1}^0 \oplus \ldots \oplus V_{\mu_s}^0
  \]
  be the generalized weight space decomposition of $V$ for $\rho$. Let
  $\rho_t$ be a continuous family of representations in
  $\Hom(\Gamma, \GL(V))$, with $\rho = \rho_0$.

  For all sufficiently small $t$, there is a $\rho_t$-invariant
  decomposition
  \[
    V_1^t \oplus \ldots \oplus V_s^t
  \]
  such that $V_i^t$ is a sum of generalized weight spaces for
  $\rho_t$, with $V_i^t$ varying continuously with $t$, and each of
  the weights associated to $V_i^t$ also varying continuously with
  $t$.
\end{prop}
\begin{proof}
  The weights vary continously as a set with multiplicity, because the
  roots of the characteristic polynomial of $\rho(\gamma)$ vary
  continuously in $\rho$ for fixed $\gamma \in \Gamma$. And, if $\mu$
  is a complex weight with multiplicity $k$, then for small $t$ there
  are complex weights $\mu_1^t, \ldots, \mu_k^t$ of $\rho_t$ (possibly
  with repeats) close to $\mu$ such that the sum of the kernels
  $\ker(\rho_\C(\gamma) - \exp(\mu_i^t(\gamma))I)^{\dim V}$ is close to
  $\ker(\rho_\C(\gamma) - \exp(\mu(\gamma))I)^{\dim V}$.
\end{proof}

\begin{definition}
  Let $\rho:H \to \SL(V)$ be a representation of a free abelian group
  $H$, and let $\Phi$ be the set of generalized weights of $\rho$. For
  any subset $\theta \subseteq \Phi$, we let $V_\theta \subseteq V$
  denote the span of the generalized weight spaces $V_\mu$ for
  $\mu \in \theta$, and we let $V_\theta^\opp \subseteq V$ denote the
  span of the generalized weight spaces $V_{\mu'}$ for
  $\mu' \in \Phi - \theta$.
\end{definition}

\subsubsection{Faces in the convex hull of the weights}

Whenever $\rho:H \to \SL(V)$ is a representation of a free abelian
group with rank $k$, we can extend any real (generalized) weight
$\mu:H \to \R$ to a homomorphism $\mu : H \ten \R \to \R$, and view it
as an element of $(\R^k)^* \simeq \R^k$.

\begin{definition}
  Let $\rho:H \to \SL(V)$ be a representation of a free abelian group,
  and let $\Phi$ be the set of generalized weights of $\rho$. We
  denote the closed convex hull of $\Phi$ in
  $(H \ten \R)^* \simeq (\R^k)^*$ by $\mc{C}(\rho)$; since $\Phi$ is a
  finite subset of $(\R^k)^*$, $\mc{C}(\rho)$ is a convex polytope in
  $(\R^k)^*$.
\end{definition}

The convex polytope $\mc{C}(\rho)$ is important for our purposes
because it tells us how to find attracting subspaces for $\rho(H)$. In
particular, there is a correspondence between the \emph{faces} of
$\mc{C}(\rho)$ and attracting subspaces of $\rho(H)$.
\begin{definition}
  Let $\rho:H \to \SL(V)$ be a represention of a free abelian group
  with generalized weight set $\Phi$. Let $F$ be a closed face of
  $\mc{C}(\rho)$. We let $\Phi(F)$ denote the set of generalized
  weights of $\rho$ lying in $F$.

  For a face $F$ of $\mc{C}(\rho)$, we write $V_F$ and $V_F^\opp$ for
  $V_{\Phi(F)}$ and $V_{\Phi(F)}^\opp$, respectively.
\end{definition}


\begin{prop}
  \label{prop:boundary_weights_determine_dynamics}
  Let $\rho:H \to \SL(V)$ be a representation of a free abelian
  group. For any divergent sequence $h_n \in H$, there is a face $F$
  of $\mc{C}(\rho)$ such that $V_F$ and $V_F^\opp$ are respectively
  attracting and repelling subspaces for $\rho(h_n)$.

  Conversely, for any face $F$ of $\mc{C}(\rho)$, $V_F$ is an
  attracting subspace for some sequence $\rho(h_n)$ with $h_n \in H$
  divergent.
\end{prop}

To prove \Cref{prop:boundary_weights_determine_dynamics}, we first
establish some estimates which will later help us show that the
convergence to the spaces $V_F$ is \emph{uniform} and
\emph{stable}. To make our estimates explicit, we choose an inner
product on $V$, which induces a norm $|| \cdot ||$ on $V$ and a smooth
metric $d_\P$ on $\P(V)$. Specifically, for any transverse subspaces
$W, W' \subset V$, we define
\[
  \angle(W, W') = \inf_{\substack{w \in W - \{0\},\\w' \in W' -
      \{0\}}} \angle(w, w'),
\]
and then take $d_\P([u], [v]) = \sin(\angle([u],[v]))$. We also choose
a norm $|\cdot|$ on $H \ten \R \simeq \R^k$.
\begin{lem}
  \label{lem:weight_norm_estimate}
  Given a representation $\rho:H \to \SL(V)$ of a free abelian group
  $H$, there exists a constant $D = D(\rho) > 0$ (varying continuously
  with $\rho$) satisfying the following: for any generalized weight
  $\mu$ of $\rho$ with nilpotence degree $\ell$, any
  $h \in H \ten \R$, and any $v$ in $V_\mu$, we have
  \[
    \frac{1}{D}(\exp(\mu(h))||v|| \le ||\rho(h)v|| \le (\exp(\mu(h))
    \cdot D|h|^{\ell-1})||v||.
  \]
\end{lem}
\begin{proof}
  For any $h \in H \ten \R$, we let $I + N_\rho(h)$ be the unipotent
  part of the (multiplicative) Jordan-Chevalley decomposition of
  $\rho_\C(h)$. That is, if we fix a generalized weight $\mu$, and let
  $\mu_\C$ be an associated complex weight, we can write
  \[
    \rho_\C(h)v = \exp(\mu_\C(h))(I + N_\rho(h))v
  \]
  for any $v \in V_{\mu_\C}$. Moreover, for any $v \in V_{\mu_\C}$, we
  have $N_\rho(h)^\ell v = 0$, where $\ell$ is the nilpotence degree
  of $\mu_\C$.

  By considering $\rho_\C(t \cdot h)$, we see that for any fixed
  $h \in H \ten \R$, $N_\rho(t \cdot h)$ is a polynomial in $t$ of
  degree at most $\dim V$, with coefficients varying continuously with
  $\rho$ and $h$, and a root at $t = 0$. Moreover, the restriction of
  $N_\rho(t \cdot h)$ to $V_{\mu_\C}$ is a polynomial in $t$ with
  degree $\ell-1$.

  Then, since $\P(H \ten \R)$ is compact, we can find $D = D(\rho)$
  (varying continuously with $\rho$) so that for any $h \in H \ten \R$
  and any $v \in V_\mu \ten \C$,
  \[
    \frac{1}{D}||v|| \le ||(I + N_\rho(h))v|| \le D \cdot |h|^{\ell-1}
    ||v||,
  \]
  where $||\cdot||$ is the norm on $V \ten \C$ induced by our norm on
  $V$.

  Then, for any $v \in V_\mu$, we have
  \begin{align*}
    ||\rho(h)v|| = ||\rho_\C(h)v|| &= ||\exp(\mu_\C(h)(I + N_\rho(h))v||\\
                                   &\le |\exp(\mu_\C(h))| \cdot D |h|^{\ell-1} ||v||\\
                                   &= \exp(\mu(h)) \cdot D |h|^{\ell-1}||v||.
  \end{align*}
  The left-hand inequality follows similarly.
\end{proof}

The lemma below gives us a uniform estimate for the amount a group
element $\rho(h)$ ``attracts'' towards a direct sum of weight spaces
of $\rho$, in terms of the weights of $\rho$.

\begin{lem}
  \label{lem:weight_gap_uniform_convergence}
  Fix a constant $M \ge 0$, and let $\rho:H \to \SL(V)$ be a
  representation of a free abelian group $H$. Let $D = D(\rho)$ be the
  constant coming from \Cref{lem:weight_norm_estimate}, and for any
  subset $\theta$ of the generalized weights $\Phi$ of $\rho$, let
  $\ell = \ell(\theta)$ be the maximum nilpotence degree of any weight
  in $\Phi - \theta$. Then, if $h \in H \ten \R$ satisfies
  \[
    \mu(h) - \mu^\opp(h) \ge M|h|
  \]
  for any $\mu \in \theta$ and $\mu^\opp \in \Phi - \theta$, we have
  \[
    \frac{d_\P(\rho(h)x, \P(V_\theta))}{d_\P(\rho(h)x,
      \P(V_\theta^\opp))} \le \frac{D^2|h|^{\ell-1}}{\exp(M|h|)} \cdot
    \frac{d_\P(x, \P(V_\theta^\opp))^{-1}}{\sin^2\angle(V_\theta,
      V_\theta^\opp)}.
  \]
  for any $x \in \P(V) - \P(V_\theta^\opp)$.
\end{lem}
\begin{proof}
  Fix a representation $\rho:H \to \SL(V)$ and a subset $\theta$ of
  the generalized weights of $\rho$, giving us a pair of complementary
  subspaces $V_\theta, V_\theta^\opp \subset V$. Let
  $x = [v] \in \P(V) - \P(V_\theta^\opp)$. We can uniquely write
  $v = v_\theta + v_\theta^\opp$ for $v_\theta \in V_\theta$ and
  $v_\theta^\opp \in V_\theta^\opp$, with $v_\theta \ne 0$. Then we
  have
  \[
    \frac{||v_\theta^\opp||}{||v||}\sin \angle(V_\theta,
    V_\theta^\opp) \le d_\P(x, \P(V_\theta)) \le
    \frac{||v_\theta^\opp||}{||v||},
  \]
  where $||\cdot||$ is the norm on $V$ induced by our choice of inner
  product. Similarly we have
  \[
    \frac{||v_\theta||}{||v||}\sin \angle(V_\theta, V_\theta^\opp) \le
    d_\P(x, \P(V_\theta^\opp)) \le \frac{||v_\theta||}{||v||}.
  \]
  So in particular, we have
  \begin{equation}
    \label{eq:subspace_dist_estimate}
    \sin(\angle(V_\theta,
    V_\theta^\opp)) \frac{||v_\theta^\opp||}{||v_\theta||} \le
    \frac{d_\P(x, \P(V_\theta))}{d_\P(x, \P(V_\theta^\opp))} \le
    \frac{1}{\sin \angle(V_\theta, V_\theta^\opp)} \cdot
    \frac{||v_\theta^\opp||}{||v_\theta||}. 
  \end{equation}
  Now let $h \in H \ten \R$ satisfy $\mu(h) - \mu^\opp(h) \ge M |h|$
  for all $\mu \in \theta$ and $\mu^\opp \in \Phi - \theta$. Since
  $V_\theta$ and $V_\theta^\opp$ are $\rho(H)$-invariant, we have
  \begin{equation}
    \label{eq:distance_norm_bound}
    \frac{d_\P(\rho(h)x, \P(V_\theta))}{d_\P(\rho(h)x, \P(V_\theta^\opp))} \le
    \frac{1}{\sin \angle(V_\theta, V_\theta^\opp)} \cdot
    \frac{||\rho(h)v_\theta^\opp||}{||\rho(h)v_\theta||}. 
  \end{equation}
  Since $v_\theta$ and $v_\theta^\opp$ are respectively linear
  combinations of elements of $V_\mu$ for $\mu \in \theta$ and
  $V_{\mu^\opp}$ for $\mu^\opp \in \Phi - \theta$, we can apply
  \Cref{lem:weight_norm_estimate} to $||\rho(h)v_\theta||$ and
  $||\rho(h)v_\theta^\opp||$ to see that
  \[
    \frac{d_\P(\rho(h)x, \P(V_\theta))}{d_\P(\rho(h)x,
      \P(V_\theta^\opp))} \le \frac{D^2|h|^{\ell-1}}{\exp(M|h|)} \cdot
    \frac{1}{\sin \angle(V_\theta, V_\theta^\opp)}
    \frac{||v_\theta^\opp||}{||v_\theta||} .
  \]
  Then, applying the left-hand inequality in
  \eqref{eq:subspace_dist_estimate} and using the fact that
  $d_\P(x, \P(V_\theta)) \le 1$, we see that
  \[
    \frac{d_\P(\rho(h)x, \P(V_\theta))}{d_\P(\rho(h)x,
      \P(V_\theta^\opp))} \le \frac{D^2|h|^{\ell-1}}{\exp(M|h|)} \cdot
    \frac{d_\P(x, \P(V_\theta^\opp))^{-1}}{\sin^2\angle(V_\theta,
      V_\theta^\opp)}.
  \]
  as required.
\end{proof}

\begin{proof}[Proof of
  \Cref{prop:boundary_weights_determine_dynamics}]
  Let $\rho:H \to \SL(V)$ be a representation of a free abelian group,
  let $\Phi$ be the set of generalized weights, and let $h_n$ be a
  divergent sequence in $H$. Up to subsequence, the sequence
  $h_n / |h_n|$ converges to some $h_\infty \in H \ten \R$ with
  $|h_\infty| = 1$.

  We can view $h_\infty$ as a linear functional on the space
  $(H \ten \R)^*$. Since $\mc{C}(\rho) \subset (H \ten \R)^*$ is a
  convex polytope, this means there is a face $F$ of $\mc{C}$ so that
  for any $\mu \in \Phi(F)$ and any $\mu^\opp \in \Phi - \Phi(F)$, we
  have $\mu(h_\infty) > \mu^\opp(h_\infty)$. Then for sufficiently
  large $n$ we also have $\mu(h_n/|h_n|) > \mu^\opp(h_n/|h_n|)$. In
  fact, since $\Phi$ is finite, there is a constant $M > 0$ such that
  $\mu(h_n) - \mu^\opp(h_n) > M|h_n|$ for every $\mu \in \Phi(F)$ and
  every $\mu^\opp \in \Phi - \Phi(F)$. Applying
  \Cref{lem:weight_gap_uniform_convergence} we see that for any
  $x \in \P(V) - \P(V_F^\opp)$, the distance
  \[
    d_\P(\rho(h_n)x, \P(V_F)) \le \frac{d_\P(\rho(h_n)x,
      \P(V_F))}{d_\P(\rho(h_n)x, \P(V_F^\opp))}
  \]
  tends to $0$ as $n \to \infty$, so $V_F$ and $V_F^\opp$ must
  respectively be attracting and repelling subspaces for $\rho(h_n)$.

  Conversely, if $F$ is any face of $\mc{C}(\rho)$, we can choose
  $h \in H \ten \R$ so that $\mu(h) > 0$ and $\mu(h) > \mu^\opp(h)$
  for any $\mu \in \Phi(F)$ and $\mu^\opp \in \Phi - \Phi(F)$. Then if
  $h_n \in H$ is any divergent sequence with $[h_n] \to [h]$ in the
  projectivization $\P(H \ten \R)$, another application of
  \Cref{lem:weight_gap_uniform_convergence} shows that $V_F$ is an
  attracting subspace for $h_n$.
\end{proof}

\subsection{Weights of peripheral subgroups in convex cocompact
  groups}

For the rest of this section, we fix a relatively hyperbolic pair
$(\Gamma, \mc{H})$, where each $H \in \mc{H}$ is virtually abelian
with rank at least 2. We also fix a representation
$\rho:\Gamma \to \SL(V)$ which is convex cocompact in $\P(V)$, and let
$\Omega \subset \P(V)$ be a properly convex domain where
$\rho(\Gamma)$ acts convex cocompactly.

Our goal now is to describe the convex polytope in $(H \ten \R)^*$
associated to the restriction of $\rho$ to each $H \in \mc{H}$, which
we can use to understand the dynamics of both $\rho(H)$ and
$\rho^m(H)$.

\begin{definition}
  For each $H \in \mc{H}$, we let $\verts{H} \subset \P(V)$ denote the
  set of vertices of $\Delta_H$.
\end{definition}

\begin{prop}
  \label{prop:vertices_weight_spaces}
  Let $H \in \mc{H}$ be a peripheral subgroup of rank $k \ge 2$, and
  let $H_0 \subseteq H$ be a finite-index free abelian
  subgroup. Consider the restriction $\rho_0 = \rho|_{H_0}$. Then, the
  convex polytope $\mc{C}(\rho_0)$ is a $k$-simplex in
  $(H_0 \ten \R)^*$, and each vertex of $\mc{C}(\rho_0)$ is a weight
  $\mu$ whose associated weight space is a vertex of $\Delta_H$.

  Moreover, every weight of $\rho_0$ which is not a vertex of
  $\mc{C}(\rho_0)$ lies in the interior of $\mc{C}(\rho_0)$.
\end{prop}
\begin{proof}
  Each vertex $v \in \verts{H}$ lies in a weight space of $\rho_0$,
  with an associated weight $\mu_v$. Let $\Phi$ denote the weights of
  $\rho_0$, and let $\Phi(\verts{H}) \subseteq \Phi$ be the set of
  weights of the form $\mu_v$ for $v \in \verts{H}$. We claim that for
  any $\mu \in \Phi - \Phi(\verts{H})$ and any $h \in H_0 \ten \R$,
  there is a vertex $v \in \verts{H}$ such that
  \[
    \mu_v(h) > \mu(h).
  \]
  Suppose for a contradiction that the claim does not hold, i.e. there
  exists $h \in H_0 \ten \R$ and $\mu \in \Phi - \Phi(\verts{H})$ such
  that $\mu(h) \ge \mu_v(h)$ for all $v \in \verts{H}$. Let
  $W_H = \spn(\Delta_H)$, and let $V'$ be the $\rho_0$-invariant
  subspace $V_\mu \oplus W_H$. Then $\rho_0$ induces a representation
  $\rho_0':H_0 \to \SL(V')$.

  Each vertex $v \in \mc{V}_H$ is an eigenspace for $\rho_0$, so the
  nilpotence degree of $\mu_v$ (viewed as a weight of $\rho_0'$) is
  1. Then, we can apply \Cref{lem:weight_gap_uniform_convergence} to
  $\rho_0'$ (with $\theta = \{\mu\}$) to see that for any divergent
  sequence $h_n \in H$ with $[h_n] \to [h]$ in $\P(H_0 \ten \R)$, and
  any fixed $x \in \P(V') - \P(W_H)$, the ratio
  \[
    \frac{d_\P(\rho(h_n)x, \P(V_\mu))}{d_\P(\rho(h_n)x, \P(W_H))}
  \]
  does not tend to infinity as $n \to \infty$. In particular this is
  true for some $x \in \Omega$, since $\P(V') \cap \Omega$ is
  relatively open and nonempty. But since $\Delta_H \subset \P(W_H)$,
  this contradicts the fact that $\dee \Delta_H$ is the full orbital
  limit set of $\rho(H_0)$ in $\Omega$.

  We have now proved our claim, which implies that any extreme point
  of the convex polytope $\mc{C}(\rho_0)$ is a weight $\mu_v$ for
  $v \in \verts{H}$. On the other hand, by
  \Cref{cor:segments_in_peripherals}, we may assume that each vertex
  $v \in \verts{H}$ is an extreme point in $\dee \Omega$, and by
  \Cref{prop:attracting_spaces_in_faces}, this means that for each
  $v \in \verts{H}$, there is a sequence $h_n \in H_0$ such that $v$
  is an attracting subspace for $\rho_0(h_n)$. Since $\rho_0(H_0)$
  acts diagonalizably on $W_H$, this implies that $\mu_v$ is an
  extreme point of $\mc{C}(\rho_0)$. The last assertion of the
  proposition follows directly from the claim.
\end{proof}

\Cref{prop:vertices_weight_spaces} tells us that we can
combinatorially identify the $k$-simplex $\Delta_H$ and the
$k$-simplex $\mc{C}(\rho_0)$ for $\rho_0 = \rho|_{H_0}$. We write this
identification explicitly:
\begin{definition}
  Let $H \in \mc{H}$, and let $H_0$ be a finite-index free abelian
  subgroup. For each face $F$ of $\Delta_H$ with vertices $\mc{V}(F)$,
  we let $\rface{F}$ denote the face of $\mc{C}(\rho_0)$ whose
  vertices are the weights $\mu_v$ for $v \in \mc{V}(F)$.
\end{definition}

\subsection{Invariant simplices in the symmetric
  power}\label{subsec:symmetric_power_simplices}

Our next step is to describe the simplices $S_H \subset \P(\Sym^mV)$
which give rise to the fibers in $\hat{\Lambda}_m$ over parabolic
points, for our EGF boundary extension $\hat{\Lambda}_m \to \bgamh$.

Let $H \in \mc{H}$ have rank $k$, and let $H_0 \subseteq H$ be a
finite-index free abelian subgroup. We let $\rho_0, \rho_0^m$
respectively denote the restrictions of $\rho, \rho^m$ to $H_0$, and
let $\Phi, \Phi^m$ denote the sets of weights of $\rho_0$ and
$\rho_0^m$. We observe the following:
\begin{lem}
  \label{lem:symmetric_polytyope_simplex}
  The convex polytope $\mc{C}(\rho_0^m)$ is the $k$-simplex
  $m\mc{C}(\rho_0)$. Moreover, for every face $\rface{F}$ of
  $\mc{C}(\rho_0)$, the weights in $\Phi^m \cap m\rface{F}$ are
  exactly the vertices of the $m$th barycentric subdivision of
  $m\rface{F}$, and each such weight has a one-dimensional generalized
  weight space.
\end{lem}
\begin{proof}
  The weights of $\rho_0^m$ are exactly the set of homomorphisms of
  the form
  \[
    \sum_{\mu \in \Phi} a_\mu \mu,
  \]
  where $a_\mu \in \N \cup \{0\}$ and $\sum a_\mu = m$. In particular,
  the set of rescaled weights $\frac{1}{m}\Phi^m$ consists entirely of
  convex combinations of weights of $\rho_0$, and contains every
  weight in $\Phi$. This (together with
  \Cref{prop:vertices_weight_spaces}) implies that $\mc{C}(\rho_0^m)$
  is a $k$-simplex.

  Further, every (rescaled) weight in the boundary of the rescaled
  simplex $\frac{1}{m}\mc{C}(\rho_0^m)$ must be a convex combination
  of weights lying in a single face of the simplex
  $\mc{C}(\rho_0)$. But \Cref{prop:vertices_weight_spaces} says that
  every weight in $\Phi \cap \dee \mc{C}(\rho_0)$ is a vertex of
  $\mc{C}(\rho_0)$. So, if $F$ is a face of the simplex $\Delta_H$
  with vertices $\mc{V}(F)$, the weights in
  $\rface{F} \cap \frac{1}{m}\Phi^m$ are exactly the convex
  combinations of the form
  \begin{equation}
    \label{eq:convex_combination_weights}
    \frac{1}{m}\sum_{v \in \mc{V}(F)} a_v \mu_v,
  \end{equation}
  where $a_v \in \N \cup \{0\}$ and $\sum a_v = m$. These are exactly
  the vertices in the $m$th barycentric subdivision of $\rface{F}$,
  and in fact each such vertex has \emph{unique} expression of the
  form \eqref{eq:convex_combination_weights}. Since each weight
  $\mu_v$ for $v \in \mc{V}(F)$ has a one-dimensional generalized
  weight space, it follows that the weights in $\Phi^m \cap mF$ do as
  well.
\end{proof}

\subsubsection{The simplices $S_H \subset \P(\Sym^mV)$}

Using \Cref{lem:symmetric_polytyope_simplex}, we can define the
vertices of the simplex $S_H$: they are exactly the weight spaces for
the weights $\mu$ lying in $\Phi^m \cap \dee \mc{C}(\rho_0^m)$.

To define $S_H$ as a subset of $\P(\Sym^mV)$, we choose lifts in
$\Sym^mV$ of each vertex of $S_H$, and then take convex
combinations. Our lifts are chosen as follows: we first pick a lift
$\tilde{v} \in V$ of each vertex $v \in \verts{H}$, so that $\Delta_H$
is the projectivization of the convex hull in $V$ of
$\{\tilde{v} : v \in \verts{H}\}$. The weight space of $\mu$ for each
$\mu \in \dee \mc{C}(\rho_0^m)$ is spanned by a unique vector in
$\Sym^mV$ of the form
\[
  \tilde{v}_\mu = \prod_{v \in \verts{H}} \tilde{v}^{a_v},
\]
where $a_v \in \N \cup \{0\}$ and $\sum a_v = m$. Then we can define
$S_H$ to be the projectivization of the convex hull in $\Sym^mV$ of
the $\tilde{v}_\mu$'s.

\subsubsection{Dynamics on the simplices $S_H$}

By definition, the vertices of $S_H$ are exactly the weight spaces for
the weights in the boundary of the simplex
$\mc{C}(\rho_0^m) \subset (H_0 \ten \R)^*$. So,
\Cref{prop:boundary_weights_determine_dynamics} immediately implies
the following:
\begin{cor}
  \label{cor:symmetric_simplex_attracting_subspace}
  Let $H \in \mc{H}$. For every divergent sequence $h_n \in H$, there
  is a face $F$ of $S_H$ which spans an attracting subspace for the
  sequence $\rho^m(h_n)$.
\end{cor}

\subsection{Dual simplices in symmetric powers}

As discussed in \Cref{sec:convex_projective_egf},
\cite{dgk2017convex}, Proposition 5.6 says that since
$\Gamma \to \SL(V)$ is convex cocompact in $\P(V)$, the dual
representation $\Gamma \to \SL(V^*)$ is convex cocompact in $\P(V^*)$,
and in fact there is a domain $\Omega \subset \P(V)$ so that $\Gamma$
acts convex cocompactly on both $\Omega$ and the dual domain
$\Omega^*$. By the work of Islam-Zimmer \cite{iz2019convex}, each
virtually abelian subgroup $H \in \mc{H}$ must act cocompactly on a
properly embedded \emph{dual} simplex $\Delta_H^* \subset
\Omega^*$. And, for each vertex $w$ of $\Delta_H^*$, the projective
hyperplane $\hyp{w}$ is a supporting hyperplane of $\Omega$ at
$\dee \Delta_H$.

\subsubsection{The simplices $S_H^* \subset \P(\Sym^mV^*)$}

For each $H \in \mc{H}$, we can define an $H$-invariant \emph{dual}
simplex $S_H^* \subset \P(\Sym^mV^*)$, by carrying out the
construction we used to find $S_H$ (but this time for the dual
representation $\rho^*:\Gamma \to \SL(V^*)$). We can describe the
relationship between the simplices $S_H$ and $S_H^*$ a little more
explicitly. For a finite-index free abelian subgroup
$H_0 \subseteq H$, we let $\rho_0^*:H_0 \to \SL(V^*)$ be the dual of
the restriction of $\rho$ to $H_0$, and similarly define
$(\rho_0^m)^*:H_0 \to \SL(V^*)$. Then the weights of $\rho_0^*$ are
the negative weights of $\rho_0$, and the weights of $(\rho_0^m)^*$
are the negative weights of $\rho_0^m$.

Suppose $\mu^m$ is a weight of $\rho_0^m$ with a one-dimensional
generalized weight space $v^m$. Then, the negative weight $-\mu^m$
also has a one-dimensional generalized weight space
$w^m \in \P(\Sym^mV^*)$, and $\hyp{w^m}$ is the hyperplane spanned by
the weight spaces of the weights in $\Phi^m - \{\mu^m\}$. In
particular, we can consider the case where $\mu^m$ is a weight lying
in the boundary of $\mc{C}(\rho_0^m)$. In this case, $v^m$ is a vertex
of $S_H$, $w^m$ is a vertex of $S_H^*$, and $\hyp{w^m}$ is a
hyperplane intersecting $S_H$ in a codimension-1 face of $S_H$.

This allows us to define a simultaneous lift of the \emph{boundaries}
of the simplices $S_H, S_H^*$ in the space of flags
$\symflags(\Sym^mV)$.
\begin{definition}
  For a peripheral subgroup $H \in \mc{H}$, we let $\dee \hat{S}_H$
  denote the set
  \[
    \dee \hat{S}_H = \{(v, w) \in \symflags(\Sym^m(V)) : v \in \dee
    S_H, w \in \dee S_H^*\}.
  \]
\end{definition}
The discussion above shows that $\dee \hat{S}_H$ is a nonempty closed
invariant subset of $\symflags(\Sym^mV)$, projecting to $\dee S_H$ and
$\dee S_H^*$ under the canonical maps
$\symflags(\Sym^mV) \to \P(\Sym^mV)$ and
$\symflags(\Sym^mV^*) \to \P(V^*)$.

\subsection{Defining the boundary set}

Using the sets $\dee \hat{S}_H$, we can define our candidate for the
EGF boundary set $\hat{\Lambda}_m \subset \symflags(\Sym^mV)$ as
follows. We let $\hat{\phi}:\hat{\Lambda}_\Omega(\Gamma) \to \bgamh$
denote the boundary extension for the EGF representation $\rho$ coming
from \Cref{thm:convex_cocompact_boundary_dynamics}. For each
$z \in \bgamh$, we define the set
$\hat{\psi}_m(z) \subset \symflags(\Sym^m(V))$ by:
\[
  \hat{\psi}_m(z) =
  \begin{cases}
    \hat{\iota}(\hat{\phi}^{-1}(z)), & z \in \conbdry(\Gamma, \mc{H})\\
    \dee\hat{S}_H, & z \in \parbdry(\Gamma, \mc{H}).
  \end{cases}
\]
We define
\[
  \hat{\Lambda}_m = \bigcup_{z \in \bgamh} \hat{\psi}_m(z),
\]
and observe that
$\hat{\iota}(\hat{\Lambda}_\Omega(\Gamma)) \subset \hat{\Lambda}_m$.

The set $\hat{\Lambda}_m$ is $\rho^m(\Gamma)$-invariant, since
$\hat{\iota}$ is $\tau_m$-equivariant and the construction of the set
$\dee \hat{S}_H$ is invariant. Ultimately we want to see that
$\hat{\Lambda}_m$ is compact, and that there is a well-defined
antipodal map $\hat{\phi}_m:\hat{\Lambda}_m \to \bgamh$ giving us our
EGF boundary extension.

\subsection{Defining the boundary extension}

Our next immediate goal is to show:
\begin{prop}
  \label{prop:symmetric_square_map}
  For distinct $z_1, z_2 \in \bgamh$, the sets
  \[
    \hat{\psi}_m(z_1), \quad \hat{\psi}_m(z_2)
  \]
  are opposite (in particular, disjoint). Consequently, the map
  $\hat{\phi}_m:\hat{\Lambda}_m \to \bgamh$ given by
  \[
    \hat{\phi}_m(\xi) = z \iff \xi \in \hat{\psi}_m(z)
  \]
  is well-defined, equivariant, surjective, and antipodal.
\end{prop}

\begin{lem}
  \label{lem:simplex_intersection_closed}
  Let $X$ be a closed subset of $\Lambda_\Omega(\Gamma)$, and let
  $\mc{H}_X \subset \mc{H}$ be the set
  $\{H \in \mc{H} : \dee \Delta_H \cap X \ne \emptyset\}$.

  Then the set
  \[
    X_{\mc{H}} = X \cup \bigcup_{H \in \mc{H}_X} \Delta_H
  \]
  is closed.
\end{lem}
\begin{proof}
  Let $x_n$ be a sequence in $X_{\mc{H}}$. By compactness of
  $\Lambda_\Omega(\Gamma)$, we can choose a subsequence so that
  $x_n \to x \in \Lambda_\Omega(\Gamma)$. We wish to show that
  $x \in X_{\mc{H}}$. Since $X$ is closed and $X \subset X_{\mc{H}}$,
  we may assume that for each $n$, we have $x_n \in \dee\Delta_{H_n}$
  for some $H_n \in \mc{H}_X$.

  Up to subsequence, the sets $\dee\Delta_{H_n}$ converge to a closed
  set $\dee\Delta_\infty$ which is a connected finite union of
  (possibly degenerate) projective simplices. We must have
  $x \in \dee \Delta_\infty \subset \Lambda_\Omega(\Gamma)$. By
  definition, $\dee \Delta_{H_n}$ intersects $X$ nontrivially, and
  since $X$ is closed we must also have
  $\dee \Delta_\infty \cap X \ne \emptyset$.

  Suppose for a contradiction that $x \notin X_{\mc{H}}$. Then in
  particular $x \notin X$. Since $\dee \Delta_\infty$ intersects $X$,
  it must contain at least two points, which means that every point in
  $\dee \Delta_\infty$ lies in a nontrivial closed projective segment
  (since $\dee \Delta_\infty$ is a connected finite union of
  projective simplices). But then by
  \Cref{cor:segments_in_peripherals},
  $\dee \Delta_\infty \subset \dee \Delta_H$ for some $H \in \mc{H}$,
  and since $\dee \Delta_\infty \cap X \ne \emptyset$ we have
  $H \in \mc{H}_X$ and therefore $x \in X_{\mc{H}}$, contradiction.
\end{proof}

\begin{prop}
  \label{prop:simplex_connected_comp}
  For each $H \in \mc{H}$, there is a connected component $C_H$ of
  \[
    \Opp(\dee S_H^*) = \{x \in \P(\Sym^mV) : x \perp w \textrm{ for
      every } w \in \dee S_H^*\}
  \]
  such that for every closed subset
  $X \subset \Lambda_\Omega(\Gamma) - \Delta_H$, $C_H$ contains the
  closure of
  \[
    \iota(X) \cup \bigcup_{H' \in \mc{H}_X} S_{H'},
  \]
  where
  $\mc{H}_X = \{H \in \mc{H} : \dee \Delta_H \cap X \ne \emptyset\}$.
\end{prop}
\begin{proof}
  Let $\verts{H}$, $\verts{H}^*$ denote the vertex sets of $\Delta_H$
  and $\Delta_H^*$, respectively. Using the convexity of $\Omega$, we
  can find lifts $\tilde{w} \in V^*$ for each vertex
  $w \in \verts{H}^*$, a continuous lift $\tilde{\Lambda} \subset V$
  of $\Lambda_\Omega(\Gamma)$, and a continuous lift
  $\tilde{\Delta}_H \subset V$ of $\Delta_H$ so that
  \begin{equation}
    \label{eq:positivity_original_v}
    \tilde{w}(\tilde{\Lambda} - \tilde{\Delta}_H) > 0
  \end{equation}
  for every $w \in \verts{H}^*$.

  The lifts $\tilde{w}$ induce lifts $\tilde{w}^m \in (\Sym^mV)^*$ of
  each vertex $w^m$ of $S_H^*$. We take the set $C_H$ to be the
  projectivization of
  \[
    \{v \in \Sym^mV : \tilde{w}^m(v) > 0 \textrm{ for all vertices }
    w^m \textrm{ of } S_H^*\}.
  \]
  Every point in $\dee S_H^*$ is the projectivization of a convex
  combination of the lifts $\tilde{w}^m$. So, $C_H$ is a connected
  component of $\P(\Sym^mV) - \bigcup_{w \in \dee(S_H^*)} \hyp{w}$.

  Now let $X \subset \Lambda_\Omega(\Gamma) - \Delta_H$ be closed and
  let $Y$ be the set
  \[
    \iota(X) \cup \bigcup_{H' \in \mc{H}_X} S_{H'}.
  \]
  We wish to show that $\overline{Y} \subset C_H$. Let $X_{\mc{H}}$ be
  the set
  \[
    X_{\mc{H}} = X \cup \bigcup_{H' \in \mc{H}_X} \Delta_{H'}.
  \]
  By \Cref{lem:simplex_intersection_closed}, we can find a
  \emph{compact} lift $\tilde{X}_{\mc{H}}$ of $X_{\mc{H}}$ in $V$ so
  that $\tilde{w}(\tilde{x}) > 0$ for every
  $\tilde{x} \in \tilde{X}_{\mc{H}}$ and every $w \in \verts{H}^*$.
  We consider the set
  \[
    \Sym^m\tilde{X}_{\mc{H}} = \{\tilde{x}^m \in \Sym^mV : \tilde{x}^m
    = \prod_{i=1}^m \tilde{x}_i \textrm{ for } \tilde{x}_i \in
    \tilde{X}_{\mc{H}}\}.
  \]
  This set is the image of the $m$-fold Cartesian product
  $(\tilde{X}_{\mc{H}})^m$ under the continuous map $V^m \to \Sym^mV$
  given by $(v_1, \ldots, v_m) \mapsto v_1\cdots v_m$, so it is
  compact. Moreover, since $\Sym^m\tilde{X}_{\mc{H}}$ contains a lift
  of every vertex of every $S_{H'}$ for $H' \in \mc{H}_X$, the
  projectivization of the convex hull of $\Sym^m\tilde{X}_{\mc{H}}$
  contains $Y$, hence $\overline{Y}$.

  But from \eqref{eq:positivity_original_v}, we know that
  $\tilde{w}^m(\tilde{x}^m) > 0$ for every vertex $w^m$ of $S_H^*$ and
  every $\tilde{x}^m \in \tilde{X}_{\mc{H}}$, so we see that $C_H$
  contains the projectivization of the convex hull of
  $\Sym^m\tilde{X}_{\mc{H}}$.
\end{proof}

\begin{proof}[Proof of \Cref{prop:symmetric_square_map}]
  Let $z_1, z_2 \in \bgamh$ be distinct. If both $z_1$ and $z_2$ are
  conical limit points, the proposition follows from the antipodality
  of the EGF boundary extension
  $\hat{\phi}:\hat{\Lambda}_\Omega(\Gamma) \to \bgamh$ and the fact
  that $\hat{\iota}$ preserves transversality. On the other hand, if
  $z_1$ is a parabolic point, this follows from
  \Cref{prop:simplex_connected_comp} (and the equivalent dual
  statement).
\end{proof}

\subsection{Dynamics on $S_H$}

We have now defined an equivariant antipodal surjective map
$\hat{\phi}_m:\hat{\Lambda}_m \to \bgamh$, but we do not yet know that
the set $\hat{\Lambda}_m$ is compact, or even that $\hat{\phi}_m$ is
continuous. However, it turns out that it is easier to verify these
two facts after proving that $\hat{\phi}_m$ has certain dynamical
properties.

\begin{lem}
  \label{lem:convergence_to_symmetric_simplex}
  For each $H \in \mc{H}$, there exists an open set
  $\hat{C}_H \subset \symflags(\Sym^mV)$ containing
  $\hat{\Lambda}_m - \dee \hat{S}_H$, such that for any infinite
  sequence $h_n \in H$ and $\xi \in \hat{C}_H$, we have
  \[
    \rho^m(h_n)\xi \to \dee \hat{S}_H.
  \]
\end{lem}
\begin{proof}
  For each $H \in \mc{H}$, we let $C_H \subset \P(\Sym^mV)$ be the set
  coming from \Cref{prop:simplex_connected_comp}. Let $h_n$ be a
  divergent sequence in some $H \in \mc{H}$, and let $H_0$ be a
  finite-index free abelian
  subgroup. \Cref{cor:symmetric_simplex_attracting_subspace} says that
  some face $F$ of $S_H$ spans an attracting subspace for
  $\rho^m(h_n)$. The corresponding repelling subspace is a direct sum
  of weight spaces for the restriction $\rho^m|_{H_0}$, so it is
  contained in $\hyp{w^m}$ for a vertex $w^m$ of the dual simplex
  $S_H^*$. So, for any $x \in C_H$, any subsequence of $\rho^m(h_n)x$
  subconverges to a point in $[\spn(F)]$. In fact, $\rho^m(h_n)x$
  subconverges to a point in $\overline{F} \subset \dee S_H$, since
  $C_H$ is $\rho^m(H)$-invariant and
  $C_H \cap \supp(F) = \overline{F}$.

  Then, we can dually define a set $C_H^* \subset \P(\Sym^mV^*)$, and
  take
  \[
    \hat{C}_H = \{(x,w) \in \symflags(\Sym^mV) : x \in C_H, w \in C_H^*\}.
  \]
\end{proof}

\subsection{Continuity and compactness}

\begin{lem}
  The set $\hat{\Lambda}_m$ is closed.
\end{lem}
\begin{proof}
  Let $(x_n, w_n)$ be a sequence in $\hat{\Lambda}_m$, and let
  $z_n = \hat{\phi}_m(x_n, w_n)$. Up to subsequence, $z_n$ converges
  to $z \in \bgamh$.

  If $z$ is a conical limit point, let $\gamma_n$ be a sequence
  limiting conically to $z$, chosen so that for any $z' \ne z$, we
  have $\gamma_n^{-1}z' \to b$ and $\lim \gamma_n^{-1}z_n = a \ne b$.

  Then $\hat{\phi}^{-1}(\gamma_n^{-1}z_n)$ converges to
  $\hat{\phi}^{-1}(a)$, and thus $\hat{\phi}^{-1}(\gamma_n^{-1}z_n)$
  lies in a fixed compact subset $X$ of
  $\Opp(\hat{\phi}^{-1}(b)) \cap \hat{\Lambda}_\Omega(\Gamma)$. By
  definition, this means that for every $n$,
  $\hat{\phi}_m^{-1}(\gamma_n^{-1}z_n)$ lies in the set
  \[
    \iota(X) \cup \bigcup_{H' \in \mc{H}_X}S_{H'},
  \]
  Arguing as in \Cref{prop:simplex_connected_comp}, we see that this
  set is compact. So by antipodality of $\hat{\phi}_m$, the sets of
  flags $\hat{\phi}_m^{-1}(\gamma_n^{-1}z_n)$ lie in a fixed compact
  subset of
  $\Opp(\hat{\phi}_m^{-1}(b)) = \Opp(\hat{\iota}(\hat{\phi}^{-1}(b)))$
  and by \Cref{prop:symmetric_power_dynamics},
  \[
    (x_n, w_n) \in \rho^m(\gamma_n)\hat{\phi}_m^{-1}(\gamma_n^{-1}z_n)
  \]
  converges to $\hat{\iota}(\hat{\phi}^{-1}(z))$.
  
  If $z$ is a parabolic point, we let $H = \Stab_\Gamma(z)$, and
  choose $h_n \in H$ so that $h_n^{-1}z_n \in K$ for a fixed compact
  $K - \{z\}$. By \Cref{prop:simplex_connected_comp}, we know that for
  all $n$, $\hat{\phi}_m^{-1}(h_n^{-1}z_n)$ lies in a fixed compact
  subset of $\hat{C}_H$. Then,
  \Cref{lem:convergence_to_symmetric_simplex} implies that
  \[
    (x_n, w_n) \in \rho^m(h_n)\hat{\phi}_m^{-1}(h_n^{-1}z_n)
  \]
  subconverges to a point in $\dee \hat{S}_H$.
\end{proof}

\begin{prop}
  The map $\hat{\phi}_m$ is continuous.
\end{prop}
\begin{proof}
  Let $(x_n, w_n)$ be a sequence in $\hat{\Lambda}_m$, converging to
  $(x, w)$ (which we know lies in $\hat{\Lambda}_m$ by the previous
  proposition). Let $z_n = \hat{\phi}_m(x_n, w_n)$, and suppose for a
  contradiction that up to subsequence $z_n \to z$ for
  $z \ne \hat{\phi}_m(x, w)$.

  \Cref{prop:simplex_connected_comp} then implies that $z_n$ lies in a
  compact subset $K \subset \bgamh$ so that the closure of
  $\hat{\phi}_m^{-1}(K)$ is opposite to $(x,w)$. This contradicts the
  fact that $(x_n, w_n)$ converges to $(x,w)$.
\end{proof}

At this point, we have shown that
$\hat{\phi}_m:\hat{\Lambda}_m \to \bgamh$ is a continuous equivariant
surjective antipodal map, and that $\hat{\Lambda}_m$ is a compact
subset of $\symflags(\Sym^mV)$. So, we can finish the proof of
\Cref{thm:symmetric_powers_boundary_dynamics} by showing:
\begin{prop}
  The map $\hat{\phi}_m:\hat{\Lambda}_m \to \bgamh$ extends the
  convergence dynamics of $\Gamma$.
\end{prop}
\begin{proof}
  We will use \Cref{prop:conical_peripheral_implies_egf}, and show
  that $\hat{\phi}_m$ has the required dynamics on conical limit
  sequences and sequences in peripheral subgroups.

  For each conical limit point $z \in \bgamh$, we define
  $\hat{C}_z \subset \symflags(\Sym^mV)$ to be the affine chart
  \[
    \hat{C}_z = \{\xi \in \symflags(\Sym^mV) : \xi \textrm{ is
      opposite to } \hat{\phi}_m^{-1}(z)\}.
  \]
  If $\gamma_n$ is a sequence limiting conically to $z$, and
  $\gamma_n^{-1}$ limits conically to $z_-$, then
  $\hat{\phi}^{-1}(z)$, $\hat{\phi}^{-1}(z_-)$ are respectively
  attracting flags for $\rho(\gamma_n)$, $\rho(\gamma_n^{-1})$. It
  follows that $\hat{\iota}(\hat{\phi}^{-1}(z))$,
  $\hat{\iota}(\hat{\phi}^{-1}(z_-))$ are respectively attracting
  flags for $\rho^m(\gamma_n)$ and $\rho^m(\gamma_n^{-1})$, so the
  required dynamics for the sequence are a consequence of
  \Cref{lem:klp_contraction_symmetry}.

  On the other hand, for each parabolic point $p$, we take $\hat{C}_p$
  to be the open set $\hat{C}_H$ considered in
  \Cref{lem:convergence_to_symmetric_simplex}, for
  $H = \Stab_\Gamma(p)$. \Cref{lem:convergence_to_symmetric_simplex}
  implies that $\hat{C}_p$ contains
  $\hat{\Lambda}_m - \hat{\phi}_m^{-1}(p)$ and that for any
  $(x, w) \in \hat{C}_p$ and any infinite sequence $h_n \in H$,
  $\rho^m(h_n)\xi$ subconverges to a point in $\dee \hat{S}_H$.
\end{proof}

\subsection{Stability}

We have now shown that the representations $\rho^m$ are all extended
geometrically finite. \Cref{thm:cusp_stable_stability} says that small
peripherally stable deformations of $\rho^m$ are also extended
geometrically finite. So we want to show the following:

\begin{prop}
  \label{prop:sym_power_peripheral_stab}
  The space $\Hom(\Gamma, \SL(\Sym^mV))$ is peripherally stable at
  $(\rho^m, \hat{\phi}_m)$.
\end{prop}

The main step in the proof is the following:
\begin{lem}
  \label{lem:symmetric_power_perturb}
  Let $H_0$ be a finite-index free abelian subgroup of some
  $H \in \mc{H}$, with $H = \Stab_\Gamma(p)$. For any open set
  $U \subset \P(\Sym^mV)$ containing $S_H$ and any $x \in C_p$, there
  exists a cofinite subset $T \subset H_0$ and an open set
  $\mc{W} \subset \Hom(\Gamma, \SL(\Sym^mV))$ containing $\rho^m$ such
  that for any $\sigma \in \mc{W}$, we have $\sigma(h) \cdot x \in U$
  for any $h \in T$.
\end{lem}
\begin{proof}
  We proceed by contradiction, and suppose that there exists a
  sequence of distinct group elements $h_n \in H_0$ and a sequence of
  representations $\sigma_n:\Gamma \to \SL(\Sym^mV)$ such that
  $\sigma_n \to \rho^m$ and $\sigma_n(h_n)x \notin U$. We let $\Phi^m$
  denote the set of generalized weights of $\rho^m|_{H_0}$, and we let
  $\Phi^m_n$ denote the generalized weights of $\sigma_n|_{H_0}$.

  We choose a norm $|\cdot|$ on $H_0 \ten \R$. Then up to subsequence
  $h_n / |h_n|$ converges to $h_\infty \in H_0 \ten \R$ with
  $|h_\infty| = 1$.

  Since $\Phi^m$ is finite, there is a face $\tilde{F}$ of the
  $k$-simplex $\mc{C}(\rho^m|_{H_0})$ and a constant $M > 0$, such
  that for every weight $\mu \in \Phi^m(F) = \Phi^m \cap \tilde{F}$,
  and every weight $\mu^\opp \in \Phi^m - \Phi(F)$, we have
  \[
    \mu(h_\infty) - \mu^\opp(h_\infty) > M.
  \]
  We let $V^m_F \subset \Sym^mV$ denote the span of the weight spaces
  of the weights in $\Phi^m(F)$; by definition $\P(V^m_F)$ is the
  projective span of a face of $S_H$.
  
  \Cref{prop:weights_deform_continuously} implies that as a set with
  multiplicity, the weights $\Phi_n^m$ converge to the weights
  $\Phi^m$. So, for each $n$, there is a subset
  $\theta_n \subset \Phi_n^m$ such that $\theta_n$ converges to
  $\Phi^m \cap \dee \mc{C}(\rho^m|_{H_0})$, and a subset
  $\theta_n(F) \subset \theta_n$ such that $\theta_n(F)$ converges to
  $\Phi^m(F)$. \Cref{prop:weights_deform_continuously} also implies
  that for sufficiently large $n$, all of the weights in $\theta_n$
  must have one-dimensional generalized weight spaces, converging to
  the vertices of $S_H$.

  This means that for each $n$, there are simplices $S_H^n$ and
  $(S_H^n)^*$, invariant under the action of $\sigma_n(H)$, such that
  $S_H^n \to S_H$ and $(S_H^n)^* \to S_H^*$. So, we can find a
  sequence of group elements $g_n \in \SL(V)$, with $g_n$ converging
  to the identity, so that $\sigma_n' = g_n \sigma_n g_n^{-1}$
  preserves the simplices $S_H$ and $S_H^*$. Moreover, the vertices of
  $S_H$ are the weight spaces $V_\mu$ of $\sigma_n'$ for
  $\mu \in \theta_n$, and the space $V^m_F$ is spanned by weight
  spaces $V_{\mu(F)}$ of $\sigma_n'$ for $\mu(F) \in \theta_n(F)$.

  Now, since $\theta_n(F)$ converges to $\Phi^m(F)$, for sufficiently
  large $n$ we must have
  \[
    \mu_n(h_\infty) - \mu_n^\opp(h_\infty) > M
  \]
  for every $\mu_n \in \theta_n(F)$ and every
  $\mu_n^\opp \in \Phi^m_n - \theta_n(F)$. This also means that for
  sufficiently large $n$ we have
  \[
    \mu_n(h_n) - \mu_n^\opp(h_n) > M|h_n|.
  \]
  Then \Cref{lem:weight_gap_uniform_convergence} implies that for
  sufficiently large $n$, $\sigma_n'(h_n)x$ lies in a small
  neighborhood of $\P(V_F)$ (here we use the fact that the constant
  $D = D(\rho)$ in \Cref{lem:weight_gap_uniform_convergence} varies
  continuously with $\rho$, and that the nilpotence degree $\ell$ of
  any weight of $\sigma_n'$ is bounded by $\dim \Sym^mV$). Moreover,
  we know that the set $C_p$ is $\sigma_n'(H)$-invariant, since the
  simplex $S_H^*$ is $\sigma_n'(H)$-invariant. Since $x$ lies in
  $C_p$, $\sigma_n'(h_n)x$ lies in an arbitrarily small neighborhood
  of $\P(V_F) \cap C_p$. By definition this intersection is a face of
  $S_H$, so for large enough $n$, $\sigma_n(h_n)x$ must lie in an
  arbitrarily small neighborhood of this face, giving a contradiction.
\end{proof}

\begin{proof}[Proof of \Cref{prop:sym_power_peripheral_stab}]
  We want to show that if $H = \Stab_\Gamma(p)$ for a parabolic point
  $p$, $K$ is a compact subset of $\hat{C}_p$, $U$ is a
  neighborhood of $\hat{S}_H$, and $T \subset H$ is a cofinite
  subset such that
  \begin{equation}
    \label{eq:cofinite_subset_in_nbhd}
    \rho^m (T) \cdot K \subset U,
  \end{equation}
  then we can find a neighborhood $\mc{W}$ of $\rho^m|_H$ in
  $\Hom(H, \SL(\Sym^mV))$ so that for any $\sigma \in \mc{W}$,
  \begin{equation}
    \label{eq:cofinite_subset_deformation}
    \sigma(T) \cdot K \subset U.
  \end{equation}

  For simplicity, we will not work in the space of flags
  $\symflags(\Sym^mV)$. Instead we will just show that that if
  (\ref{eq:cofinite_subset_in_nbhd}) holds for a compact
  $K \subset C_p \subset \P(\Sym^mV)$ and an open neighborhood $U$ of
  $S_H$ in $\P(\Sym^mV)$, then (\ref{eq:cofinite_subset_deformation})
  holds also.

  Fix a finite-index free abelian subgroup $H_0 \subseteq H$. It
  suffices to show that we can choose an open
  $\mc{W} \subset \Hom(\Gamma, \SL(\Sym^mV))$ so that
  \[
    \sigma(T \cap H_0) \cdot K \subset U
  \]
  for all $\sigma \in \mc{W}$. Using
  \Cref{lem:symmetric_power_perturb}, we can find a cofinite set
  $T' \subset H_0$ and an open set
  $\mc{W} \subset \Hom(\Gamma, \SL(\Sym^mV))$ so that for all
  $\sigma \in \mc{W}'$, we have
  \[
    \sigma(T') \cdot K \subset U.
  \]
  But then since $(T \cap H_0) - T'$ is finite, we can just shrink
  $\mc{W}$ to get the desired result.
\end{proof}



