\section{Introduction}

Let $\Sigma = \Sigma_{g,n}$ be the compact oriented surface of genus $g$
with boundary $\partial\Sigma$ which has $n\geq 1$ components
denoted $\partial_1\Sigma,...,\partial_n\Sigma$.
Fix a basepoint $x_0$ on $\Sigma$ and let $\pi = \pi_1(\Sigma,x_0)$ denote its fundamental group.
Let $\HomeoS$ be the group of 
orientation-preserving homeomorphisms of $\Sigma$
which fixes $\partial\Sigma$ pointwise.
Define the {\em mapping class group\/} of $\Sigma$:
\[ \Gamma : = \pi_0\big(\HomeoS\big).\]
Alternatively, 
choose basepoints $x_i\in\partial_i\Sigma$ and paths  from $x_0$ to $x_i$.
Define $\pi_1(\partial_i\Sigma)$ as the cyclic subgroup of $\pi$ 
corresponding to $\pi_1\big(\partial_i\Sigma,x_i\big)$
and determined by the paths between $x_0$ and $x_i$.
Let $\AutS$ denote the subgroup of $\Aut(\pi)$ 
which preserves both the conjugacy classes of the 
subgroups $\pi_1\big(\partial_i(\Sigma)\big)$ 
and the orientation of $\Sigma$.
Then $\Gamma$ is  isomorphic to the image of $\AutS$ under the quotient
homomorphism
\begin{equation*}
\Aut(\pi) \longrightarrow \Out(\pi) := \Aut(\pi)/\Inn(\pi).
\end{equation*}


Let $G$ be an algebraic group over $\R$.
Then the set of homomorphisms $\pi \to G$ 
enjoys the the structure of an $\R$-algebraic set
denoted $\Hom(\pi,G)$.
Choose conjugacy classes $\CC_i \subset G$
for each $i=1,\dots,n$ and let $\CC := \{\CC_1,\dots,\CC_n\}$.
Denote by $\HomC$ the subset of $\Hom(\pi,G)$ comprising homomorphisms which send 
$\pi_1(\partial_i\Sigma)$ to $\CC_i$
 
The group $\Inn(G)$ of inner  automorphisms of $G$
acts on $\HomC$ by composition. 
Denote the resulting {\em relative character variety\/} by:
\[
\M_\CC(G) := \HomC/\Inn(G).
\]

The group $\AutS$ acts on $\pi$, 
and hence  on $\Hom_\CC(\pi,G)$ by composition.
Furthermore, the action descends to a $\Gamma$-action on $\MCC(G)$. 
The moduli space $\MCC(G)$ has an invariant dense open subset 
$\MCCo$ which is a smooth manifold. 
If, for example, $G$ is $\R$-reductive (see \cite{Go84, Go97}), 
this subset has an $\Gamma$-invariant symplectic structure $\Omega$.
In particular, 
$\MCCo$  admits a natural smooth $\Gamma$-invariant measure $\mu$.


This paper is part of the general program to understand the dynamics of the
action of $\Gamma$ and automorphism groups of free groups
on character varieties when the Lie group is compact.
Suppose that $K$ is a compact Lie group.
In \cite{Go97}, Goldman conjectures that $\Gamma$ acts ergodically on 
$\MCC(K)$ and shows this to be the case when $K$ is locally a product of 
$\SU(2)$'s and $\U(1)$'s.  
In \cite{PX02a} and \cite{PX03a}, the conjecture is proved for $g \geq 2$, and also established for almost all boundary classes when $g=1=n$.  In \cite{GX11}, Goldman and Xia offer an alternate and simpler proof for the case of $K = \SU(2)$.  



In this paper, we consider the case $g = 1 = n$ with $K = \SUthree$.  
Then $\Gamma \cong \SL(2,\Z)$  (see \cite{FM12}). 
In this special case, $\MCC(K)$ is explicitly described by the 
{\em commutator map\/}
\begin{align*}
K \times K &\xrightarrow{~\kappa~}  K \\
(a,b) &\longmapsto [a,b] := aba\inv b\inv.
\end{align*}
Indeed, $\MCC(K) \cong \kappa\inv(\CC)/\Inn(K)$, 
for each conjugacy class $\CC \subset K$.


\begin{thm}\label{thm:main}
Let $K = \SU(3)$ and $\surf=\surf_{1,1}$. 
The $\Gamma$-action is ergodic on $\MCC(K)$ with 
respect to the measure $\mu$.
\end{thm}

We prove this theorem along the lines of the main results in \cite{GX11}.  If $\rho \in \HomC$, we denote its $\Inn(G)$-equivalence class as $[\rho]$.  Similarly if $S \subseteq \HomC$, then the corresponding set of $\Inn(G)$-equivalence classes is denoted by $[S]$.
A simple closed curve $\alpha$ on $\Sigma$
defines a function 
\begin{align*}
\MCC(K) &\xrightarrow{~\f_\alpha~} \C \\
[\rho] &\longmapsto \Tr\big(\rho(\alpha)\big).
\end{align*} 
The symplectic structure $\Omega$ together with the real and imaginary parts of $\f_\alpha$ give rise to Hamiltonian flows.  
The ring-theoretical results in \cite{La07} imply the algebra of Hamiltonian
vector fields is {\em infinitesimally\/} transitive.
It follows that the group generated by these flows is {\em locally\/} transitive, and hence, ergodic.

Depending on the choice of $\alpha$, these Hamiltonian flows preserve the sets $[\H(a, b)]$ or $[\H'(a,b)]$ defined in Section~\ref{subsec:DH}.
On the other hand, $\Gamma$ contains the Dehn twist $\tau_\alpha$ along $\alpha$.  The $\tau_\alpha$ action also preserves $[\H(a,b)]$ and is ergodic in almost all $[\H(a,b)]$. 
It follows that a $\mu$-measurable function is
invariant under $\tau_\alpha$ if and only if it 
is invariant under the Hamiltonian flows associated with $\f_\alpha$.
By local transitivity, such a measurable function is almost everywhere constant.



\subsection*{Acknowledgements}
Goldman and Lawton were partially supported by U.S. National Science Foundation grants: DMS 1107452, 1107263, 1107367 1309376 ``RNMS: Geometric structures And Representation varieties" (the GEAR Network).  Lawton acknowledges support from U.S. National Science Foundation grants: DMS 1309376, and also DMS 0932078000 while he was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2015 semester.  He was also partially supported by a Simons Foundation Collaboration grant. Xia was partially supported by the Ministry of Science and Technology Taiwan with grants 103-2115-M-006-007-MY2, 105-2115-M-006-006, 106-2115-M-006-008 and 107-2115-M-006-009 and the National Center for Theoretical Sciences, Taiwan.

\section*{Notation and terminology}

Let $G$ be a group.  Denote the inner automorphism induced by $A\in G$ by:
\begin{align*}
G &\xrightarrow{~\Inn(A)~} G \\
B  &\longmapsto ABA\inv
\end{align*}
The set of conjugacy classes in $G$ equals the quotient $G/\Inn(G)$,
and we denote the image of a subset $S\subseteq G$ under the quotient map
by $[S]$.

Denote the {\em centralizer\/} of $A$ in $G$ by:
\[ G_A := \Fix\big(\Inn(A)\big) \subgroup G, \]
where $\Fix(S)$ is the set of fixed points of $S \subseteq G$.

Define the commutator map of $G$:
\begin{align*}
G \times G &\xrightarrow{~\kappa~} G \\
(A,B) &\longmapsto [A,B] := ABA^{-1}B^{-1}. \end{align*}


Suppose further that $G$ is a
Lie %
group  with Lie algebra $\g$,
and denote the adjoint representation of $G$ on $\g$ by $\Ad$.
Identify $\g$ with the Lie algebra of {\em right-invariant vector fields\/} on $G$;
then for any right-invariant vector field $X \in \g$,
the element $\Ad(a)(X)$ equals the image of $X$ under left-multiplication
by $a$.
Denote the {\em centralizer\/} of $a$ in $\g$ by:
\[ \g_a := \Fix\big(\Ad(a)\big) \subseteq \g. \]


Denote the trace of a matrix $a$ by $\Tr(a)$
and the $\lambda$-eigenspace of a matrix $a$ by $\Eig_\lambda(a)$,
for a scalar $\lambda\in\C$.

The notation $K$ and $\fk$ are reserved for compact Lie groups and its Lie algebra, respectively.


If $(M,\Omega)$ is a symplectic manifold,  and
$M \xrightarrow{~f~} \R$ is a smooth function,
denote its {\em Hamiltonian vector field\/} by $\Ham(f)$.
We denote the tangent space to a smooth manifold $M$ at a point $p\in M$
by $T_p M$.

When we say a set is {\em closed}, we mean it to be closed in the
classical topology.







\section{Character varieties and the mapping class group}\label{sec:char}
We fix a basepoint on the boundary of $\surf : = \surf_{1,1}$.
The fundamental group $\pi := \pi_1(\surf)$ is isomorphic to the rank 2 free group $\Ftwo$, generated
by homotopy classes of
oriented based loops $\alpha$ and $\beta$.
We often do not distinguish elements in $\pi$ from corresponding oriented based loops on $\surf$.

We write
$$
\pi = \la \alpha, \beta, \sigma | \kappa(\alpha, \beta) = \sigma \ra,
$$
where $\sigma$ is the boundary element.
In this way, we have
$$
R := \Hom(\pi, K) \cong K \times K \ \ \ \text{ and } \ \ \ \M := R/K,
$$
where the $K$-action is by conjugation.

In our case, we have only one boundary circle and we let
$\CC \subseteq K$ be a conjugacy class and $c \in \CC$.  Then the relative representation variety and character variety are
$$
R_c : = \Hom_\CC(\pi,K) := \kappa^{-1}(c) \ \text{ and } \ \M_c := R_c(K)/K_c.
$$
Again, the $K_c$-action is by conjugation.
In this way, a representation $\rho \in R_c$ corresponds to $(a,b) \in K \times K$ such that $\kappa(a,b) = c$.
Notice that $\M_c$ is usually and equivalently defined as
$$
\M_c = \kappa^{-1}(\CC)/K.
$$
The space $\M_c$ has a natural symplectic structure $\Omega$ \cite{Go84, Go97}.

The diffeomorphism group of $\Sigma$ (fixing the boundary, hence, also the basepoint) acts on $\pi$ and this action descends to a $\Gamma$-action on $\pi$, fixing the conjugacy class of $\sigma$.
This further induces an action:
\begin{align*}
\M_c \times \Gamma &\longrightarrow \M_c \\
([\rho], \gamma) &\longmapsto [\rho \circ \gamma].
\end{align*}
The $\Gamma$-action leaves $\Omega$ and $\mu$ invariant \cite{Go97}.
For any oriented simple closed curve $\alpha$ on $\surf$, denote by $\tau_\alpha$ the Dehn twist along $\alpha$.
The mapping class group $\Gamma$ %
contains all
Dehn twists; %
indeed the Dehn twists {\em generate\/} $\Gamma$
(although we do not need this fact).
Denote by $\Scal$ the set of homotopy classes of oriented simple closed curves on $\Sigma$.

\section{Compact Lie groups}
This section reviews well known facts that are used in the proofs.
{\em Generic elements\/} are introduced; these are regular elements
which are dense in their maximal tori,
and provide nontrivial dynamics.

\subsection{Regularity}

Suppose $M$ is an irreducible algebraic set over $\R$ or $\C$ and $M^s \subset M$ its singular locus.
Then $U  = M \setminus M^s$ is a smooth manifold and Zariski dense in $M$.
The smooth structure on $U$ gives rise to the Lebesgue measure class on $U$ and on $M$,
by assigning $M^s$ to be a null set.
We shall always mean this class, which coincides with the measure class discussed in the introduction \cite{Hu95}.

Let $G$ be a linear
semisimple algebraic group over $\C$ of rank $r$
and $K < G$ a maximal compact subgroup.
The corresponding Lie algebras are denoted by $\fk$ and $\g$, respectively.

Recall that an element $a \in K$ is {\bf regular} if $K_a$ has dimension $r$.
In general, $\dim(K_a) \ge r$, and $K_a$  contains a maximal torus of $K$.
An element $a \in K$ is regular if and only if $K_a$ is a maximal torus (that is, a Cartan subgroup) in $K$.

Recall that an action on a topological space is {\em minimal\/} if every orbit is dense.
If $a\in K$, denote the Haar measure on $K_a$ by
$\mu_{K_a}$ and the pushforward  $(L_b)_*\mu_{a}$ under left-multiplication
$K_a \xrightarrow{~L_b~} bK_a$ by $\mu_{ba}$.

\subsection{Genericity}
In general regularity is too weak a notion for dynamical complexity.
We introduce a notion of {\em genericity\/} which is more useful for
constructing nontrivial dynamics.

Let $(a,b)\in K\times K$.
Then the cyclic group $\la a\ra$ acts on the left-coset $b K_a$ by:
\[
b\zeta \xmapsto{~a^n~}  b\zeta a^n \]
where $n\in\Z$ and $\zeta\in K_a$.

\begin{prop}\label{prop:irrational rotation}
Let $a\in K$ be a regular element.
For any $b\in K$,
the following conditions are equivalent:
\begin{itemize}
\item the cyclic group $\la a \ra < K_a$ is Zariski dense in $K_a$;
\item the cyclic group $\la a \ra < K_a$ is dense in $K_a$;
\item the action of $\la a\ra$ on $bK_a$ is minimal;
\item the action of $\la a\ra$ on $(bK_a,\mu_{ba})$ is ergodic.
\end{itemize}
\end{prop}
In this case, we say that $a$ is {\em generic.\/}
The proof of Proposition~\ref{prop:irrational rotation} uses standard facts about compact abelian Lie groups,
such as:
\begin{lem}\label{lem:Density}
A cyclic subgroup of $K_a$ is dense in the classical topology if and only if it is dense in the Zariski topology.
\end{lem}
\begin{proof}
Any set that is classically dense is also Zariski dense since the Zariski topology is coarser than the classical topology. We now show the converse.
Clearly the cyclic group $\la a\ra \subset K_a$, and its Zariski-closure is also an abelian subgroup.
Its closure in the classical topology
\[ H :=\overline{\la a\ra} \] is a closed abelian subgroup of $K_a$.
Now every compact linear group is Zariski-closed
(See Onishchik-Vinberg~\cite{OV90}, \S 4.4, Theorem~5, pp.133--134).  Hence $H$ is Zariski closed in $K_a$.  Since ${\la a\ra}$ is Zariski dense in $K_a$ and $H \supseteq {\la a \ra}$, $H=K_a$.


\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:irrational rotation}]
The proof now follows from Lemma~\ref{lem:Density} and
the fact that dense subgroups of the torus act minimally
(see Katok-Hasselblatt \cite [\S 1.4 (p.28) ]{KH95})
and ergodically (\cite[Proposition 4.2.2 (p.147)]{KH95},
or Walters \cite[Theorem 1.9 (p.30)]{Wa82}.
\end{proof}


\section{Infinitesimal transitivity and Hamiltonian flows}\label{sec:IT}
In this section, we let $G$ be a semi-simple complex algebraic Lie group.
Let $M$ be a symplectic manifold and $M \xrightarrow{~f~} \R$ a smooth function.  
Denote by $\Ham(f)$ the associated Hamiltonian vector field.

\begin{defin}
Let $M$ be a manifold and $\Fm$ be a set of
real smooth $\R$-functions on $M$ such that at $x \in M$, the
differentials $df(x)$, for $f\in\Fm$, span the cotangent space
$T_x^*(M)$. Then $\Fm$ is said to be infinitesimally transitive at $x$.  $\Fm$ is infinitesimally transitive on $M$ if
$\Fm$ is infinitesimally transitive at all $x \in M$.
\end{defin}

\begin{prop}\label{prop:Ham}
Let $M$ be a connected symplectic manifold and $\Fm$ be infinitesimally transitive on $M$.
Then the group $\mH$ generated by the Hamiltonian flows
$\Ham(f)$ of the vector fields $\Ham(f)$, for $f\in\Fm$, acts
transitively on $M$.
\end{prop}
\begin{proof}
See Lemma 3.2 in \cite{GX11}.
\end{proof}




We now briefly review results of Goldman~\cite{Go86},
describing the flows generated by
the Hamiltonian vector fields by {\em simple\/}  closed curves on $\Sigma$.
In this case, the local flow of this vector field on $\M_c$
lifts to a flow on the representation variety $R_c$.
Furthermore this flow admits a simple description~\cite{Go86}
as follows.
\subsection{Invariant functions and flows on groups}
Let $G$ be a semi-simple complex Lie group with Lie algebra $\g$.
Then the adjoint representation $\Ad$ preserves a nondegenerate symmetric
bilinear form $\langle \_ , \_ \rangle$ on $\g$. In the case $G=\SL(n,\C)$, this is
\begin{equation*}
\langle X, Y\rangle := \Tr (X Y).
\end{equation*}

Let $G\xrightarrow{\f} \R$ be a function invariant under the inner automorphisms $\Inn(G)$.
Following \cite{Go86}, we describe how
$\f$ determines a way to associate to every element $x\in G$
a one-parameter subgroup
\begin{equation*}
\zeta^t(x) \, = \, \exp\big( t F(x) \big)
\end{equation*}
centralizing $x$.
Given  $\f$, define  its {\em variation function\/}
$ G \xrightarrow{F} \g$ by:
\begin{equation*}
\langle F(x), \upsilon \rangle = \frac{d}{dt}\bigg|_{t=0}
\f \big( x \exp(t\upsilon)\big)
\end{equation*}
for all $\upsilon\in\g$.
Invariance of $\f$ under $\Inn(G)$ implies that $F$ is $G$-equivariant:
\begin{equation*}
F( g x g^{-1}) = \Ad(g) F(x).
\end{equation*}
Taking $g = x$ implies that the one-parameter subgroup
\begin{equation}\label{eq:oneparameter}
\zeta^t(x) := \exp(t F(x))
\end{equation}
lies in the centralizer of $x\in G$.


Intrinsically, $F(x)\in\g$ is dual (by  $\langle \_, \_\rangle$) to the
element of $\g^*$ corresponding to
the left-invariant 1-form  on $G$ extending the covector
$df(x) \in T^*_x(G)$.

\subsection{Invariant functions and centralizing one-parameter subgroups}

\label{ssec:hamilton}
Recall that $\Scal$ denotes the set of homotopy classes of oriented simple closed curves on $\Sigma$.
If $\alpha \in \Scal$ is an oriented homotopy class of based loops,
then $\f_\alpha$, the {\em trace function\/} of $\alpha$, is defined as:
\begin{align*}
\Hom(\pi,G) & \xrightarrow{~\f_\alpha~} \C \\
\rho & \longmapsto  \Tr\big(\rho(\alpha)\big).
\end{align*}
Since the function $G \xrightarrow{~\Tr~} \C$ is $\Inn(G)$-invariant,
$\f_\alpha$ defines a $\C$-valued function (also denoted by $\f_\alpha$) on $\M_c$.
Let 
$$\f_\alpha^R = \Re(\f_\alpha),  \ \ \ \f_\alpha^I = \Im(\f_\alpha).$$
Then $\f_\alpha^R$ and $\f_\alpha^I$ define $\R$-valued functions on $\M_c$.  







Let $\alpha \in \Scal$ and $\Sigma | \alpha$ denote the compact surface 
obtained by {\em splitting\/} $\Sigma$ along $\alpha$. 
The two components $\alpha_\pm$ of $\partial \Sigma | \alpha$ corresponding to $\alpha$
are the preimages of $\alpha\subset\surf$ under the quotient mapping $\Sigma|\alpha \longrightarrow\surf$.
The original surface $\Sigma$ may be reconstructed as a quotient
space under the identification of $\alpha_-$ with $\alpha_+$.

The fundamental group $\pi$ can be
reconstructed from the fundamental group $\pi_1(\Sigma | \alpha)$ as
an HNN-extension:
\begin{equation}\label{eq:hnn}
\pi \;\cong\;   \bigg(\pi_1(\Sigma | \alpha) \amalg
\langle\beta\rangle \bigg)
\bigg/
\bigg(\beta \alpha_- \beta^{-1} = \alpha_+ \bigg).
\end{equation}
A representation $\rho$ of $\pi$ is determined by:
\begin{itemize}
\item the restriction $\rho'$ of $\rho$ to the subgroup
$\pi_1(\Sigma|\alpha)\subset\pi$, and
\item the value $\beta' = \rho(\beta)$
\end{itemize}
which satisfies:
\begin{equation}\label{eq:hnnrep}
\beta'  \rho'(\alpha_-) \beta'^{-1} = \rho'(\alpha_+).
\end{equation}
Furthermore any pair $(\rho',\beta')$ where $\rho'$ is a representation of
$\pi_1(\Sigma|\alpha)$ and $\beta'\in G$ satisfies \eqref{eq:hnnrep}
determines a representation $\rho$ of $\pi$.

Let $\f = \f_\alpha^R$ or $\f = \f_\alpha^I$.  
Define the {\em twist flow\/} $\xi_\alpha^t$, associated with $\f$ on $\Hom(\pi,\SU(3))$:
\begin{equation}\label{eq:hamtwist1}
\xi_\alpha^t(\rho):\gamma \longmapsto \begin{cases}  \rho(\gamma)
& \text{if}~ \gamma \in \pi_1(\Sigma|\alpha) \\
\rho(\beta) \zeta^t\big(\rho(\alpha_-)\big)
& \text{if}~ \gamma = \beta. \end{cases}
\end{equation}
where $\zeta^t$ is defined in \eqref{eq:oneparameter}.
This flow covers the flow generated
by $\Ham(\f)$ on $\M_c$
(See \cite{Go86}).

\subsection{Infinitesimal transitivity}
Let $\XX$ be the Geometric Invariant Theory quotient of  $\Hom\big(\Ftwo,\SL(3,\C)\big)$ by 
$\Inn\big(\SL(3,\C)\big)$.
Choose $\alpha, \beta \in \Scal$ as in Section \ref{ssec:hamilton} to correspond to curves with geometric
intersection number $1$ (or equivalently a free basis of $\Ftwo$).
Let
\[ \Gg := 
\{\alpha,\beta,\alpha \beta,\alpha\beta\inv,\alpha\beta\alpha\inv\beta\inv\}\subset \Scal \]
and 
$\Gg\inv := \{\gamma\inv : \gamma\in\Gg \}$. 
Let  \[\Fm_{\alpha,\beta} := \{ \f_\gamma : \gamma\in\Gg\cup \Gg\inv\}. \]

\begin{thm}[Lawton~\cite{La06,La07}]\label{thm:top of SU(3) rep}
The coordinate ring $\C[\XX]$ is generated by $\Fm_{\alpha,\beta}$.
\end{thm}
Since $\Tr(A\inv) = \overline{\Tr(A)}$ for $A\in\SUthree$,
the set $\Fm_{\alpha,\beta}$ is invariant under complex conjugation.




The relative $\SUthree$-character variety $\M_c$ of $\Ftwo$ embeds in $\XX$ as a real semialgebraic
subset. 
The regular functions on this $\R$-semialgebraic set are the real and imaginary parts 
of the restrictions to $\M_c$ of the regular functions on $\XX$.
\begin{cor}\label{prop:IT g=1}
The set 
\[ 
\Fm_{\alpha,\beta}^\R := \{\f_\gamma^R, \f_\gamma^I : \gamma \in \Scal_{\alpha,\beta}\} \]
is infinitesimally transitive on $\M_c$.  
\end{cor}

\begin{proof}
Given the remarks preceding this corollary, the result follows from Theorem \ref{thm:top of SU(3) rep} and  \cite[Lemma 3.1]{GX11}.
\end{proof}

















\section{The Dehn twists}\label{ssec:Dehn}
Let  $\alpha \in \Scal$ and $\tau_\alpha\in \Gamma$ be the corresponding Dehn twist.
The fundamental group $\pi$ can be
reconstructed from the fundamental group $\pi_1(\Sigma | \alpha)$ as
an HNN-extension as in \eqref{eq:hnn}.
Then $\tau_\alpha$ induces the automorphism
$(\tau_\alpha)_*\in\Aut(\pi)$ defined by:

\begin{equation*}
(\tau_\alpha)_*:\gamma \longmapsto
\begin{cases}  \gamma
& \text{if}~ \gamma \in \pi_1(\Sigma|\alpha) \\
\gamma  \alpha
& \text{if}~ \gamma = \beta. \end{cases}
\end{equation*}
This further induces the map $(\tau_\alpha)^*$ on $\Hom(\pi,G)$ mapping
$\rho$ to:
\begin{equation}\label{eq:dehnnonsep}
(\rho)(\tau_\alpha)^*:\gamma \longmapsto
\begin{cases}  \rho(\gamma)
& \text{if}~ \gamma \in \pi_1(\Sigma|\alpha) \\
\rho(\gamma)  \rho(\alpha)
& \text{if}~ \gamma = \beta. \end{cases}
\end{equation}
(See \cite{Go86}).

\subsection{Dehn twists and Hamiltonian twist flows}\label{subsec:DH}


Let $a := \rho(\alpha)$ and $b = \rho(\beta).$
Then
\begin{align*}
R_c & = \{\rho \in \Hom(\pi, K) : \kappa(\rho(\alpha), \rho(\beta)) = c\} \\ & = \{(a,b) \in K \times K : \kappa(a,b) = c\}.
\end{align*}
Let
$$\H(a,b) := \{a\}\times bK_a, \ \ \ \ \H'(a,b) := aK_b \times \{b\}$$

\begin{prop}\label{prop:flow}
If $(a,b) \in R_c$, then
$\H(a,b), \H'(a,b) \subseteq R_c.$
\end{prop}
\begin{proof}
Suppose $t \in K_b$.  Then $at = ta$ and $t^{-1}a^{-1} = a^{-1}t^{-1}$.
Then
$$
k(a,bt) = a(bt)a^{-1}(bt)^{-1} = abta^{-1}t^{-1}b^{-1} = aba^{-1}b^{-1} = k(a,b) = c.
$$
Hence $(a,bt) \in R_c$.  The proof of $aK_b \times \{b\} \subseteq R_c$ is similar.
\end{proof}
\begin{prop}\label{prop: H-orbit}
If $(a,b) \in R_c$,
then the Hamiltonian flows of the vector fields $\Ham(\f_\alpha^R)$ and  $\Ham(\f_\alpha^I)$ preserve $[\H(a,b)]$.  Similarly, $\Ham(\f_\beta^R)$ and $\Ham(\f_\beta^I)$ preserve $[\H'(a,b)]$.
\end{prop}
\begin{proof}
This follows from \eqref{eq:hamtwist1} and by exchanging $\alpha$ and $\beta$.
\end{proof}






\begin{cor}\label{cor:generic1}
If $a$ is generic, then $\la \tau_\alpha\ra$ acts ergodically on $\H(a,b)$.
If $b$ is generic, then $\la \tau_\beta\ra$ acts ergodically on $\H'(a,b)$.
\end{cor}

\begin{proof}
By \eqref{eq:dehnnonsep},
$\tau_\alpha(a,b) \in \H(a,b)$ and $\tau_\beta(a,b) \in \H'(a,b)$
The corollary then follows from Proposition \ref{prop:irrational rotation}.
\end{proof}























\section{The case of $K = \SU(3)$}
For the rest of this paper, we denote $\omega := e^{2\pi i/3}$ and the identity transformation by $\Id$.
In this section we fix $K = \SU(3)$.  

The classification of conjugacy classes of $K$ can be described in terms of the {\em trace\/} function
\[
K \xrightarrow{~\Tr~} \C. \]
Let $\Delta = \Tr(K)$.

If $\zeta_1,\zeta_2,\zeta_3\in\C$ are the eigenvalues of $a\in K$,
then they satisfy
\begin{equation}\label{eq:Eigenvalues}
\vert \zeta_1\vert = \vert \zeta_2\vert = \vert \zeta_3\vert = 1 \text{~and~} \zeta_1 \zeta_2 \zeta_3 = 1. \end{equation}
 The coefficients of the character polynomial $\chi_a$ are:
\begin{align*}
1 & = 1\\
\zeta_1 + \zeta_2 + \zeta_3 &  = \Tr(a) \\
\zeta_2\zeta_3 + \zeta_3\zeta_1 + \zeta_1\zeta_2 & = \overline{\Tr(a)} \\
\zeta_1 \zeta_2\zeta_3 & = 1.
\end{align*}
Therefore the  characteristic polynomial is:
\[
\chi_A(\lambda) \  = \
\lambda^3 \  - \ z\, \lambda^2 \ + \ \bar{z} \,\lambda \  - 1\]
where $z = \Tr(a)\in\C$.
Furthermore \eqref{eq:Eigenvalues} is equivalent to the condition:
\[
\vert z \vert^4 -  8\mathsf{Re} (z^3)  + 18 \vert z\vert^2  - 27 \le 0 \]
and this real polynomial condition exactly describes the image $\Delta \subseteq \C$.


\begin{figure}\label{fig:Delta}
 \includegraphics[scale=0.5]{SUthreeTraces.pdf}
 \caption{$\Delta$ the traces in $\SU(3)$.}
 \end{figure}

The traces of central elements are the vertices $3, 3\omega, 3\bar{\omega}$ of
$\Delta$.
The trace of a regular element of order three is the center $0$ of $\Delta$.
The traces of elements of order two are $-1, -\omega, -\bar{\omega}$,
the midpoints of the edges of $\partial \Delta$.

\begin{prop}\label{prop:Sard}
The map $\Tr$ is a local submersion at almost all points of $\Delta$.
\end{prop}
\begin{proof}
The map $\Tr$ is smooth.  Hence, by Sard's theorem, almost all points of $\C$ are regular values of $\Tr$ \cite[\S 1.7]{GP10}.  Hence $\Tr$ is a local submersion at almost all points of $\Delta$.
\end{proof}
\begin{rem}
It is not difficult to show that $\Tr$ has full rank in the interior of $\Delta$, but we only need Proposition \ref{prop:Sard}.  For a general discussion of $\Delta$ and Weyl chambers, see \cite{DK00}.
\end{rem}

\begin{prop} \label{cor:GenericityConullInDelta}
The image $\Tr(N)$ of the subset $N \subseteq K$ of generic elements
is conull in  $\Delta$.
\end{prop}
\begin{proof}
Let
$$
U = \{(\alpha, \beta) \in \R \times \R : \alpha, \beta, \frac{\alpha}{\beta} \in \R \setminus \Q\}
$$
Let $\Delta'\subset\Delta$ be the image of $U$ under the mapping
\begin{align*}
\R \times \R &\longrightarrow \C \\
(\alpha, \beta) &\longmapsto e^{2\pi i\alpha} + e^{2\pi i\beta} + e^{-2\pi i(\alpha +\beta)}
\end{align*}
Then $\Delta'$ is conull in $\Delta$ and $\Tr^{-1}(\Delta') \subseteq N$.
\end{proof}










\section{Central fibers of $\kappa$}
Again in this section, we fix $K = \SU(3)$.
\subsection{The abelian representations}\label{sec:ab}
The fiber $R_\I = \kappa\inv(\I)$ consists of commuting pairs $(a,b)$.
In this case, $a$ and $b$ lie in a maximal torus $\bT^2$.  Hence $\M_\I \cong (\bT^2 \times \bT^2)/W$, where $W$ is the Weyl group of $K$ acting diagonally (see \cite{FL14} for more discussion of these abelian character varieties)

\subsection{The non-abelian cases}\label{sec:nab}
\begin{prop}\label{prop:single point}
If $k = \omega\I$ where $\omega\neq 1$, then $\M_k$ consists of a single point.
Specifically, if $(a,b)\in R_k$, then there exists $g\in K$ such that
\[
a = g a_0 g\inv , \qquad b = g b_0 g\inv \]
where
\begin{equation}\label{eq:A0B0}
a_0 := \bmatrix 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \endbmatrix, \qquad
b_0 := \bmatrix 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \endbmatrix. \end{equation}
\end{prop}
\begin{proof}
Suppose $(a,b)\in R_k$, that is,
\begin{equation} \label{eq:CommOmega}
a b a\inv b\inv = \omega\I.  \end{equation}
We first prove:
\begin{lem}\label{lem:OrderThree}
$a^3 = b^3 = \Id$ \end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:OrderThree}]
By \eqref{eq:CommOmega} and taking traces,
\begin{equation*}
\Tr(a)  = \Tr(\omega b a b\inv) = \omega \Tr(a)    \ \text{ and } \
\Tr(a\inv)  = \Tr(\omega b\inv a\inv b) = \omega \Tr(a^{-1}).
\end{equation*}
Hence $\omega\neq 1$ implies that $\Tr(a) = \Tr(a\inv) = 0$.
Now apply the Cayley-Hamilton theorem:
\[ a^3 -  \Id = a^3 - \Tr(a) a^2 + \Tr(a\inv) a - \Det(a) \Id = 0. \]
The same argument applied to
$b   = \omega a\inv b a$ implies $b^3=\Id$, as claimed. \end{proof}
Returning to the proof of Proposition~\ref{prop:single point},
Lemma~\ref{lem:OrderThree} implies that
$a = \Inn(g) a_0$ for some $g\in K$, since $a\neq \I$.

We claim that $b = \Inn(g) b_0$.
For convenience assume that $a = a_0$;
then we show that \eqref{eq:CommOmega} implies that
$b$ is the permutation matrix $b_0$ defined in \eqref{eq:A0B0}.


Recall that $\Eig_\lambda(a)$ is the $\lambda$-eigenspace of $a$.
We claim that \eqref{eq:CommOmega} implies that
\begin{equation}\label{eq:PermuteEig}
b \big( \Eig_\lambda (a) \big) \ = \ \Eig_{\omega\lambda} (a). \end{equation}
To see this, rewrite \eqref{eq:CommOmega} as:
\begin{equation}\label{eq:ABBA}
a b \ = \ \omega ba  \end{equation}
Suppose that $v\in  \Eig_\lambda (a)$, that is,
\[
a v = \lambda v \]
so applying \eqref{eq:ABBA},
\[
a (b v) \ = \ \omega b a v \ = \ \omega\lambda (b v) \]
whence $b v \in \Eig_{\omega\lambda} (a)$, as claimed.

Since $a$ is the diagonal matrix $a_0$ defined by \eqref{eq:A0B0},
the lines
$\Eig_1(a), \Eig_\omega(a),$ and $\Eig_{\bar\omega}(a)$ are the three coordinate lines in $\C^3$.
Thus \eqref{eq:PermuteEig} implies that $b = b_0$,
concluding the proof of Proposition~\ref{prop:single point}.
\end{proof}

In particular, both $a$ and $b$ have order $3$.
Since $\kappa(a,b)$ has order 3 and is central, 
the subgroup $\langle a, b\rangle \subset K$ is a nonabelian group of order $27$,
a nontrivial central extension of $\Z/3 \oplus \Z/3$ by $\Z/3$.













\section{Ergodicity}
Let $K$ be any compact Lie group.
Each tangent space $T_a K$ identifies with the Lie algebra $\fk$ of {\em right-invariant vector fields:\/}
namely, a tangent vector $v\in T_a K$ identifies with the right-invariant vector $X \in \fk$ such that $X(a) = v$.  In this way, the differential of the commutator map $\kappa$ at
$(a,b) \in K \times K$ identifies with the linear map (see \cite{Go84, Go20, PX02a}):
\begin{align*}
 \D\kappa_{(a,b)} : \fk \oplus \fk & \longrightarrow \fk
\\
 (X,Y) &\longmapsto \Ad(ba)\big((\Ad(b^{-1}) - \I\big)X  \\
 & \qquad \qquad + \big(\I - \Ad(a^{-1})\big)Y.
\end{align*}
From this formula, the following proposition holds:
\begin{prop}\label{prop:C-singular}
$\kappa$ is
submersive %
at $(a,b)$ if and only if $\fk_a \cap \fk_b = 0$.
\end{prop}
For the rest of this section, let $K = \SU(3)$ which comes with the standard representation $\Pi$ on $\C^3$.  An element in $(a,b) \in R_c$ corresponds to a representation $\rho$ of $\pi$ (see Section \ref{sec:char}).  Hence $\rho \circ \Pi$ is a representation of $\pi$.  Denote by $\M_c^i \subseteq \M_c$ the subspace of irreducible representation classes.


\subsection{Relation to the moduli space of $K$-bundles}  If we fix a complex structure on $\Sigma$, then we can construct the coarse moduli space $\N^{ss}$ of semi-stable parabolic $K$-bundles \cite{BR89} on $\Sigma$, endowed with a complex structure.  $\N^{ss}$ contains the subspace $\N^s$ of stable parabolic $K$-bundles.





\begin{prop}\label{prop:connected}
The set of smooth points of $\M_c$ is a connected manifold.
\end{prop}
\begin{proof}
There is a homeomorphsim $\M_c \cong \N^{ss}$, restricting to a diffeomorphism $\M_c^i \cong \N^s$ (Theorem 1, \cite{BR89}).
The moduli space $\N^{ss}$ is irreducible and contains $\N^s$ as an open subvariety (Theorem II,  $\M_c$ \cite{BR89}).  Hence $\N^s$ is open and connected.  Hence $\M_c^i$ is open and connected. Proposition \ref{prop:C-singular} implies that
a point $[\rho] = [(a,b)] \in \M_c$ is a smooth point if and only if $\rho$ is irreducible, i.e. $\M_c^i$ is also the set of smooth points of $\M_c$.  The proposition follows.
\end{proof}



For almost every conjugacy class $c$,
the action $\M_c \times \Gamma \to \M_c$ is ergodic \cite{PX02a}.
This section proves that this is true for {\em all} $c$.


Let $ c \in K$.  Up to conjugation,
\[
c = \bmatrix c_1 & 0 & 0 \\ 0 & c_2 & 0 \\ 0 & 0 & c_3  \endbmatrix.
\]
For $(a,b) \in R_c$, we have the natural map
$$
\iota : K_a \lto \H(a,b), \ \ \ \iota(t) = (a,bt).
$$




Let $P_2 : K \times K \lto K$ be the projection to the second factor.
An element $\rho \in R_c$ corresponds to a pair $(a,b) \in K \times K$.  Let
$$\T = \Tr \circ P_2 : R_c \lto \Delta \ \ \text{ and } \ \  T := \iota \circ \T : K_a \lto \Delta.$$





\begin{prop}\label{prop:submersion}
Let $c \not\in Z(K)$.
Then $(a,b) \in R_c$ exists such that:
\begin{enumerate}
\item $(a,b)$ is a smooth point;
\item $T$ is a submersion at $(a,b)$.
\end{enumerate}
\end{prop}

\begin{proof}
Let $(a,b) \in R_c$ such that
\begin{equation}\label{eqn:ab}
a = \bmatrix 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0  \endbmatrix, \ \ \
b = \bmatrix b_1 & 0 & 0 \\ 0 & b_2 & 0 \\ 0 & 0 & b_3  \endbmatrix.
\end{equation}
Then
\begin{equation}
c = \kappa(a,b) = \bmatrix \frac{b_2}{b_3} & 0 & 0 \\ 0 & \frac{b_3}{b_1} & 0 \\ 0 & 0 & \frac{b_1}{b_2}  \endbmatrix.
\end{equation}
This formula for $c$ implies that $\kappa$ is onto $K$.
\begin{rem}
A result of Goto \cite{Go49} states that for any compact semi-simple Lie group K, $K \times K \xrightarrow{~\kappa~} K$ is surjective.  Hence $R_c \neq \emptyset$ for all $c \in K$.
\end{rem}


Note that $a$ is regular.  For (1), there are three cases for $b \in Z(K)$, $b_1 = b_2 \neq b_3$ (and its permutation variation) and $b$ being regular.


If $b \in Z(K)$, then $\kappa(a,b) = \I = c$ and this violates our hypothesis of $c \neq Z(K)$.

If $b$ is regular, then $t \in \fk_b$ implies
\begin{equation}\label{eq:t}
t = \bmatrix t_1 & 0 & 0 \\ 0 & t_2 & 0 \\ 0 & 0 & t_3  \endbmatrix,  \ \ \ t_1 + t_2 + t_3 = 0.
\end{equation}
Then
$$
(\Ad(a) - \I)t = \bmatrix t_2 - t_1 & 0 & 0 \\ 0 & t_3 - t_2 & 0 \\ 0 & 0 & t_1 - t_3  \endbmatrix.
$$
Hence if $t \in \fk_a$, then $(\Ad(a) - \I)t = 0$.  This implies $\fk_a \cap \fk_b = 0$.
Hence, by Proposition \ref{prop:C-singular}, we conclude that $\kappa$ is regular at $(a,b)$.

If $b_1 = b_2 \neq b_3$ and $t \in \fk_b$, then
$$
t = \bmatrix t_{11} & t_{12} & 0 \\ t_{21} & t_{22} & 0 \\ 0 & 0 & t_{33}  \endbmatrix, \ \ \ t_1 + t_2 + t_3 = 0.
$$
Then
$$
(\Ad(a) - \I)t = \bmatrix t_{22} - t_{11} & -t_{12} & t_{21} \\ -t_{21} & t_{33} - t_{22} & 0 \\ t_{12} & 0 & t_{11} - t_{33}  \endbmatrix.
$$
Hence if $t \in \fk_a$, then $(\Ad(a) - \I)t = 0$ This implies $t = 0$.  Hence $\fk_a \cap \fk_b = 0$.
By Proposition \ref{prop:C-singular}, $\kappa$ is regular at $(a,b)$.  We conclude that in all cases $(a,b) \in R_c$ is a smooth point.

Notice that $\H(a,b) \subseteq R_c$.
We consider $T$ restricted to $\H(a,b)$.
Let
$$
p = \bmatrix 1 & \omega^2 & \omega \\ 1 & \omega & \omega^2 \\ 1 & 1 & 1  \endbmatrix \ \ \text{ and } \ \
q = \bmatrix 1 & 0 & 0 \\ 0 & \omega^2 & 0 \\ 0 & 0 & \omega  \endbmatrix.
$$
Then
$a = p q p^{-1}$.  The element $t \in K_a$ has the form $t = p t_b p^{-1}$, where $t_b \in K_b$ is diagonal. 
Then
$$
T(t) = \Tr(bt) = \frac{1}{3} \Tr(b) \Tr(t).
$$
By Proposition \ref{prop:Sard}, $T$ is a local submersion for all almost all $t \in \Delta$ unless $\Tr(b)=0$.  However $\Tr(b) = 0$ implies $c \in Z(K)$, a contradiction.  Hence $\Tr(b) \neq 0$ and $T$ is a local submersion for almost all $t$.

\end{proof}
\begin{cor}
$\T$ is a local submersion for almost all points in $\H(a,b)$.
\end{cor}
\begin{proof}
Since $\D T = \D \T \circ \D \iota,$  $\D T_{t}$ being surjective implies $\D \T_{(a,bt)}$ is surjective.
\end{proof}

\begin{cor}\label{cor:generic2}
There is a conull set $V \subset R_c$ such that $b$ is generic for almost all $(a,b) \in V$.
\end{cor}
\begin{proof}
The subset containing points at which a map is locally submersive is Zariski open.
Hence,
by Proposition~\ref{prop:submersion}, there exists smooth and Zariski open $V \subseteq R_c$ such that $T|_V$ is a submersion.  Let $Q \subseteq R_c$ be the smooth part.  By Proposition~\ref{prop:connected}, $Q$ is connected, hence, irreducible.  Since Zariski open subset of a smooth irreducible variety is conull in the Lebesgue class, $V$ is conull.  The map $T|_V$ is a fibration over an open domain of $\Delta$.
The corollary follows from Proposition~\ref{cor:GenericityConullInDelta}.
\end{proof}


\begin{cor}\label{cor:GH}
Suppose $c \not\in Z(K)$.  Suppose $\beta \in \Scal$ and $\phi : \M_c \lto \R$ is a $\mu$-measurable function.  If $\phi$ is $\tau_\beta$-invariant, then $\phi$ 
is $\Ham(\f_\beta^R)$-invariant and $\Ham(\f_\beta^I)$-invariant.
\end{cor}

\begin{proof}
By Proposition~\ref{prop:submersion}, $R_c$ contains a smooth point $(a,b)$ with $b$ generic.
By Proposition~\ref{prop:connected} and Corollary~\ref{cor:generic2},
$b \in K$ is generic for almost all $(a,b)\in R_c$.  Hence $b \in K$ is generic for almost all $[(a,b)] \in \M_c$.  By Proposition~\ref{prop:irrational rotation}, \ref{prop:flow} and Corollary~\ref{cor:generic1}, $\tau_\beta$-orbit is dense in $\H'(a,b)$ and $\phi$ is $\Ham(\f_\beta^R)$-invariant and $\Ham(\f_\beta^I)$-invariant.
\end{proof}

With the notation we have adopted, we restate Theorem \ref{thm:main} as:
\begin{thm}
The action $\M_c \times \Gamma \to \M_c$ is ergodic.
\end{thm}

\begin{proof}
Suppose that $c = \I$.
Identifying $\R^2$ with its dual $(\R^2)^*$, the group $\SL(2,\Z)$ has the standard dual linear action on $\R^2$ which induces the diagonal action on $\bT^2 \times \bT^2$.  This $\SL(2,\Z)$-action is known to be ergodic because $\SL(2,\Z)$ contains hyperbolic elements, meaning that the eigenvalues of these elements do not have absolute value $1$ \cite[\S 4]{BS15}.

There is an isomorphism \cite{FM12}
$
\iota : \Gamma \stackrel{\cong}{\lto} \SL(2,\Z).
$
By Section \ref{sec:ab}, $\M_c \cong (\bT^2 \times \bT^2)/W$.  The $\Gamma$-action on $\M_c$ lifts to an action on $\bT^2 \times \bT^2$.
Moreover, this $\Gamma$-action is equivariant with respect to $\iota$.  Hence the $\Gamma$-action on $\M_c$ is ergodic.



Suppose that $c \in Z(K)$ and $c \neq \I$, then $\M_c$ is a single point by Proposition~\ref{prop:single point} and the statement is trivially true.





Suppose $c \not\in Z(K)$.
Recall that $\mH$ is the group generated by all Hamiltonian flows $\Ham(\f_\beta^R)$ and $\Ham(\f_\beta^I)$ where $\beta \in \Scal$.  Let $\phi : \M_c \lto \R$ be a $\Gamma$-invariant $\mu$-measurable function.  By Corollary \ref{cor:GH}, $\phi$ is $\mH$-invariant. By Corollary~\ref{prop:IT g=1} and Proposition~\ref{prop:Ham}, $\phi$ is constant almost everywhere.  Our theorem follows.
\end{proof}




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