\section{Introduction}
In this paper, we study continuous (or Borel) actions of non-Archimedean TSI Polish groups on Polish spaces. A Polish group is {\bf non-Archimedean} if it is homeomorphic to a closed subgroup of $S_\infty$, the Polish group of permutations of the natural numbers. And a Polish group is {\bf TSI} if it has a metric $d$ that is complete and compatible with the topology and also two-sided invariant (i.e. $d(a, b) = d(ca, cb) = d(ac, bc)$ for every $a, b, c$ in the group). However the characterization we will use in this paper is that the non-Archimedean TSI Polish groups are exactly the closed subgroups of product groups $\prod_{n=1}^\infty \Delta_n$ for countable groups $\Delta_n$. Such groups have a natural tree structure called the {\bf group tree}, which we study in order to understand the orbit equivalence relations $E^G_X$ induced by continuous actions of such Polish groups $G$ on Polish spaces $X$.

An equivalence relation $E$ on a Polish space $X$ has {\bf potential complexity} $\Gamma$ iff there is a Polish topology $\sigma$ on $X$ such that $E$ has complexity $\Gamma$ as a subset of $X \times X$ with the product topology. The complexity classes that we will consider in this paper will form a hierarchy, and in fact are well-ordered. This hierarchy is used to compare the complexities of various equivalence relations.

Another way to measure the complexity of equivalence relation is by using Borel reductions. Given two equivalence relations $E$ and $F$ on Polish spaces $X$ and $Y$, we say that $E$ is {\bf Borel reducible} to $F$, denoted $E \le_B F$, iff there exists a Borel function $f : X \rightarrow Y$ such that $x_1 E x_2$ iff $f(x_1) F f(x_2)$ for every $x_1, x_2 \in X$. In the case that $f$ can be taken to be continuous, we say that $E$ is {\bf continuously reducible} to $F$. For the complexity classes $\Gamma$ that we will consider, it will be obvious that if $E \le_B F$ and $F$ is potentially $\Gamma$, then $E$ is potentially $\Gamma$. Thus Borel reducibility is a finer measure of complexity than potential complexity. In fact, it is strictly finer -- the entire field of study of countable Borel equivalence relations (CBERs) consists of $\mathbf{\Sigma}^0_2$ equivalence relations, and yet there are many equivalence relations that are pairwise incomparable with respect to Borel reducibility. We will develop some techniques in this paper to better understand actions of non-Archimedean TSI Polish groups in terms of these two measures.

This paper is part of a larger project to understand the potential Borel complexity of orbit equivalence relations induced by various classes of non-Archimedean Polish groups (See \cite{HKL_1998}, \cite{Ding_Gao_2017}, \cite{Shani_1}, \cite{Shani_2}, and \cite{Clemens_Coskey}).

\subsection{The Potential Complexity Spectrum}
In \cite{HKL_1998}, a thorough analysis was performed for actions of non-Archimedean Polish groups with Borel orbit equivalence relations, completely characterizing the possible potential Borel complexity classes. They came up with the following list:


\begin{equation}\label{eq:complexity_spectrum}
\mathbf{\Delta}^0_1, \mathbf{\Pi}^0_1, \mathbf{\Sigma}^0_2, \mathbf{\Pi}^0_n, D(\mathbf{\Pi}^0_n) \text{ ($n \ge 3$)}, \bigoplus_{\alpha < \lambda} \mathbf{\Pi}^0_\alpha, \mathbf{\Pi}^0_\lambda, \mathbf{\Sigma}^0_{\lambda+1}, \mathbf{\Pi}^0_{\lambda+n}, D(\mathbf{\Pi}^0_{\lambda+n}) \text{ ($n \ge 2$, $\lambda$ limit)}. \tag{$*$}
\end{equation}
Here, an equivalence relation $E$ is $D(\mathbf{\Pi}^0_\alpha)$ iff it is the difference of two $\mathbf{\Pi}^0_\alpha$ sets, and $\bigoplus_{\alpha < \lambda} \mathbf{\Pi}^0_\alpha$ iff there is a partition $X = \bigsqcup_{n \in \omega} X_n$ of the domain of $E$ into Borel sets such that for every $n$, there is some $\alpha < \lambda$ such that $E \cap (X_n \times X_n)$ is $\mathbf{\Pi}^0_\alpha$.

It's straightforward to check that this list is linearly-ordered in the natural way. For example, starting from the beginning, we have:
\[\mathbf{\Delta}^0_1 \subseteq \mathbf{\Pi}^0_1 \subseteq \mathbf{\Sigma}^0_2 \subseteq \mathbf{\Pi}^0_3 \subseteq D(\mathbf{\Pi}^0_3) \subseteq \mathbf{\Pi}^0_4 \subseteq D(\mathbf{\Pi}^0_4) \subseteq ... \]
The authors of \cite{HKL_1998} proved that this list of potential complexities was exhaustive for actions of non-Archimedean Polish groups with Borel equivalence relations, in the sense that for such an orbit equivalence relation $E$, and any ordinal $\alpha$, if $E$ is potentially $\mathbf{\Pi}^0_\alpha$ (resp. potentially $\mathbf{\Sigma}^0_\alpha$), then there is a complexity $\Gamma$ in $(*)$ such that $E$ is potentially $\Gamma$, and $\Gamma \subseteq \mathbf{\Pi}^0_\alpha$ (resp. $\Gamma \subseteq \mathbf{\Sigma}^0_\alpha)$.

For a non-Archimedean Polish group $G$, we will let the {\bf potential Borel complexity spectrum} of $G$ to be exactly the complexity classes $\Gamma$ in $(*)$ such that there is a Polish $G$-space $X$ such that $E^G_X$ is potentially $\Gamma$, but not potentially $\Gamma'$ for any complexity class $\Gamma'$ appearing earlier in the list.

This raises a family of interesting questions:
\begin{question}\label{q:complexity_spectrum}
Given a non-Archimedean Polish group $G$, what is its potential Borel complexity spectrum?
\end{question}

By exploiting the natural tree structure of the non-Archimedean TSI Polish groups, we can answer this for a large class of grousp.

\begin{restatable*}{theorem}{tsicomplexityspectrumthm}\label{th:tsi_complexity_spectrum}
For any non-Archimedean TSI Polish group $G$, the potential Borel complexity spectrum of the group $G^\omega$ is an initial segment of $(*)$.
\end{restatable*}

From this we can conclude that if $G$ is a non-Archimedean TSI Polish group, and $G^\omega$ induces Borel orbit equivalence relations of arbitrarily high complexity, then every complexity class in $(*)$ is in the potential Borel complexity spectrum of $G^\omega$. In this case, we say that this group has {\bf full potential Borel complexity spectrum}. It was proven in \cite{HKL_1998} that $S_\infty$ has full potential Borel complexity spectrum. Using Solecki's results on tame abelian product groups (see \cite{Solecki_1995}), we can give more examples of such groups.

\begin{restatable*}{corollary}{tsicomplexityspectrumcor}\label{cor:tsi_complexity_spectrum}
The groups $\mathbb{Z}^\omega$, $(\mathbb{Z}_p^{<\omega})^\omega$ for prime $p$, and $F_2^\omega$ all have full potential Borel complexity spectrum.
\end{restatable*}

\subsection{Universal Actions}
In addition to characterizing the potential complexity spectrum of $S_\infty$, the authors of \cite{HKL_1998} defined, for each complexity class, a single action of $S_\infty$ belonging to that class, that is also universal for actions of $S_\infty$ belonging to that class. Given a Polish group $G$ and complexity class $\Gamma$, we say that an equivalence relation $E$ is {\bf universal for orbit equivalence relations induced by $G$ of potential complexity $\Gamma$} if $E$ is induced by a continuous action of $G$ and is potentially $\Gamma$, and for any other such equivalence relation $F$, we have $F \le_B E$. So in other words, they showed that for every complexity class $\Gamma$ in $(*)$, there is an equivalence relation that universal for orbit equivalence relations induced by $S_\infty$ of potential complexity $\Gamma$.
We may ask this in general:
\begin{question}
For a given Polish group $G$ and complexity class $\Gamma$, is there an equivalence relation that is universal for orbit equivalence relations induced by $G$ of potential complexity $\Gamma$?
\end{question}

For the same class of groups as before, we give a positive answer to this question.
\begin{restatable*}{theorem}{tsiuniversalactionsthm}\label{th:tsi_universal_actions}
For any non-Archimedean TSI Polish group $G$, and any complexity class $\Gamma$ in $(*)$, there is an equivalence relation that is universal for orbit equivalence relations induced by actions of $G^\omega$ of potential complexity $\Gamma$.
\end{restatable*}
The constructions are similar in spirit to the iterated ``group jump" constructions of \cite{Clemens_Coskey}.

We can generalize this definition in a natural way. Let $\mathcal{C}$ be a class of Polish groups, such as the non-Archimedean TSI Polish groups. We say that an equvivalence relation $E$ is {\bf universal for potentially $\Gamma$ orbit equivalence relations induced by $\mathcal{C}$-actions} if $E$ is potentially $\Gamma$, induced by a continuous action of some group in $\mathcal{C}$, and for any such equivalence relation $F$, we have $F \le_B E$. In \cite{HKL_1998}, it is shown for every complexity $\Gamma$ in $(*)$ that there is an equivalence relation that is universal for potentially $\Gamma$ orbit equivalence relations induced by actions of non-Archimedean Polish groups. We can show that this is also true when $\mathcal{C}$ is the class of non-Archimedean TSI and non-Archimedean abelian Polish groups.

\begin{restatable*}{corollary}{tsiuniversalactionscor}\label{cor:tsi_universal_actions}
For $\mathcal{C}$ equal to the non-Archimedean abelian Polish groups, or the non-Archimedean TSI Polish groups, and for any $\Gamma$ in $(*)$, there is an equivalence relation that is universal for potentially $\Gamma$ orbit equivalence relations induced by $\mathcal{C}$-actions.
\end{restatable*}

\subsection{Classification by non-Archimedean TSI actions}

Given a class $\mathcal{C}$ of Polish groups, we say that an equivalence relation $E$ is {\bf classifiable by $\mathcal{C}$-actions} if there is a Polish group $G$ in $\mathcal{C}$ and a Polish $G$-space $X$ such that $E \le_B E^G_X$. 
In general, showing that an equivalence relation $E$ is not classifiable by $\mathcal{C}$-actions is a difficult task, and usually involves identifying some kind of ``topological obstruction". 
Given two equivalence relation $E$ and $F$ living on $X$ and $Y$ respectively, we say that $E$ is {\bf generically $F$-ergodic} if for any Borel homomorphism $f : X \rightarrow Y$ there is a comeager subset $C \subseteq X$ such that $f[C]$ is contained in a single $F$-class. 
As an example of a topological obstruction, if an equivalence relation $E$ is \emph{turbulent} as in \cite{Hjorth_2000}, then $E$ is generally-ergodic with respect to any orbit equivalence relation that is induced by a continuous action of a non-Archimedean Polish group. 
If we also have that all orbits of $E$ are meager, then $E$ is not classifiable by non-Archimedean actions. A natural question is to ask the following:

\begin{question}
What topological obstructions exist for classification by non-Archimedean TSI actions?
\end{question}

In \cite{Allison_Panagio_2020}, a topological obstruction was given for classification by general TSI actions in the case that $E$ is an orbit equivalence relation, called the \emph{generic semi-connectnedness of the balanced graph}. If we are only concerned with the non-Archimedean TSI Polish groups, the topological obstruction is very simple.

\begin{restatable*}{theorem}{topobstructionthm}\label{th:top_obstruction}
An equivalence relation $E$ is generically ergodic with respect to any orbit equivalence relation $E^G_X$ with $G$ non-Archimedean TSI if and only if $E$ is generically $E_\infty$-ergodic.
\end{restatable*}

The equivalence relation $E_\infty$ is defined as the shift action of $F_2$, the free group on two-generators, on $2^{F_2}$. However the property we will be exploiting is the fact that it is the maximal countable Borel equivalence relation under Borel reducibility (see e.g.  \cite[Theorem 7.3.8]{Gao_2009}).

In \cite{Clemens_Coskey}, Clemens and Coskey define a jump operator $E \mapsto E^{[\Delta]}$ on equivalence relations for every countable group $\Delta$, and show that in the case that $E$ is countable it is classified by a non-Archimedean CLI (complete left-invariant) Polish group. As it was not known if classification by CLI Polish groups implied classification by TSI Polish groups, they asked if these relations are classified by TSI Polish groups. We can apply Theorem \ref{th:top_obstruction} to conclude that in many cases, such as $E_0^{[\mathbb{Z}]}$, it is not classifiable by non-Archimedean TSI Polish groups.

\begin{restatable*}{corollary}{ccapplication}\label{cor:cc_application}
If $E$ is generically ergodic and countable and $\Delta$ is infinite, then $E^{[\Delta]}$ is not classifiable by non-Archimedean TSI Polish groups.
\end{restatable*}

This result was extended in \cite{Allison_Panagio_2020} to show that these relations are not classifiable by general TSI Polish groups.

If we follow the same line of argument with respect to $E_0$-ergodicity, we get the following result:
\begin{restatable*}{theorem}{topobstructionthmtwo}\label{th:top_obstruction2}
An equivalence relation $E$ is generically ergodic with respect to any orbit equivalence relation $E^G_X$ with $G$ non-Archimedean abelian if and only if $E$ is generically $E_0$-ergodic.
\end{restatable*}

We can also analyze classification by $\mathcal{C}$-actions of a given Borel complexity. Given a complexity $\Gamma$ from $(*)$, we say that an equivalence relation $E$ is {\bf classifiable by $\mathcal{C}$-actions of potential complexity $\Gamma$} if there is some Polish group $G$ from $\mathcal{C}$ and a Polish $G$-space $X$ that is potentially $\Gamma$ such that $E \le E^G_X$.

In \cite[Theorem 6.1]{Ding_Gao_2017}, Ding and Gao proved that if $E$ is an essentially countable equivalence relation and $E \le_B E^G_X$ where $G$ is non-Archimedean and abelian, then $E \le_B E_0$. Recall that $E_0$ is the equivalence relation on $2^\omega$ such that $x E_0 y$ iff there is some $n$ such that for every $k \ge n$, $x(k) = y(k)$.

We will be able to generalize this result by characterizing the equivalence relations classifiable by $\mathcal{C}$-actions of potential complexity $\Gamma$ when $\mathcal{C}$ is the class of non-Archimedean TSI Polish groups, or the non-Archimedean abelian Polish groups. 

\begin{restatable*}{corollary}{classificationcor}\label{cor:classification}
Let $E$ be an equivalence relation that is Borel-reducible to a $\mathbf{\Pi}^0_3$ orbit equivalence relation (for example, if $E \le_B =^+)$. If $E$ is instead classifiable by non-Archimedean TSI actions, then $E \le_B E_\infty^\omega$. If $E$ is also classifiable by non-Archimedean abelian actions, then $E \le_B E_0^\omega$.
\end{restatable*}


\subsection{Structure of the Paper}

In Section 2, we cover some basic definitions and results about Borel reductions and changes of topology that we will need throughout the paper. In Section 3, we review Hjorth's Scott-like analysis of Polish group actions as well as some other facts about Borel complexity of orbit equivalence relations that we will need.  In Section 4, we define a family of universal equivalence relations, for each TSI non-Archimedean Polish group and every potential Borel complexity class.  In Section 5, we prove all of the results stated in the introduction.












\section{Preliminaries}

Given a sequence $(\Delta_i)_{i \in \omega}$ of countable groups, as in \cite{Solecki_1995}, define $\Delta^n$ to be the product $\prod_{i < n} \Delta_i$, treating $\Gamma^0$ as the trivial group. 
A {\bf group tree} is a subset $T \subseteq \bigcup_{n \in \omega} \Delta^n$ for some sequence $(\Delta_i)_{i \in \omega}$ such that $T$ is closed under initial segments (i.e. $T$ is a tree), and for every $n$, $T^n := T \cap \Delta^n$ is a subgroup of $\Delta^n$.

Given a tuple $a \in T$, let $\ell(a)$ be the length of $a$. Given two tuples $a$ and $b$ of the same length $n$, let $a \cdot b$ be the group product of $a$ and $b$ as elements of $T^n$. We will also let $a^{-1}$ refer to the inverse of $a$ as an element of $T^n$, and $1^n$ refer to the identity element of $T^n$.

Given a tuple $a \in T$, let $T_a$ be the set of $b$ such that $a \smallfrown b \in T$, where $a \smallfrown b$ is the concatination of $a$ and $b$.
Similarly, let $T_a^k$ be the set of $b \in T_a$ of length exactly $k$.

Given a group tree $T \subseteq \bigcup_{n \in \omega} \Delta^n$, define $G_T$ to be the closed (and hence Polish) subgroup of $\prod_{n \in \omega} \Delta_n$ consisting of the branches through $T$.
For every $a \in T$, let
\[V^T_a := \{g \in G_T \mid a \sqsubseteq g\}.\]
Then $\{V^T_{1^n} \mid n \in \omega\}$ forms a countable local basis for the identity of $G_T$ of open normal subgroups, and $\{V^T_a \mid a \in T\}$ is a countable basis. We will assume that $T$ is {\bf pruned}, i.e. every sequence in $T$ can be properly extended in $T$, so that each $V^T_a$ is nonempty.

Letting $\mathcal{I}$ be a countable basis for the standard topology on $\mathbb{R}$ (such as the collection of open intervals with rational endpoints), and $f$ a partial functon from $T$ to $\mathcal{I}$, let $\mathcal{N}_f$ be the set of $\mathbf{x} \in \mathbb{R}^T$ such that for every $a$ in the domain of $f$, $\mathbf{x}(a) \in f(a)$. Let $\mathcal{F}$ be the countable set of finite suppont functions $T \rightarrow \mathcal{I}$. Then it is easy to see that the $\mathcal{N}_f$ for $f \in \mathcal{F}$ form a countable basis for a Polish topology on $\mathbb{R}^T$.  

Any non-Archimedean TSI Polish group $G$ is isomorphic to a closed subgroup of $\prod_{i = 0}^\infty \Delta_i$ for a countable sequence $(\Delta_i)_{i \in \omega}$ of countable groups. This fact is found in \cite[Lemma 12.1]{Hjorth_Kechris_1997} (also appeared independently in \cite{Megrelishvili_Shlossberg_2013} and \cite{Gao_Xuan_2014}). This tells us that we can write $G$ as $G_T$ for some group tree $T$. Thus in order to understand the non-Archimedean TSI Polish groups, it is enough to understand the groups $G_T$.

We will be concerned with the Polish space $\mathbb{R}^T \times T$ with the product topology of the topology we just described on $\mathbb{R}^T$ and the discrete topology on the countable set $T$.
Furthermore, there is a natural action of $G_T$ on $\mathbb{R}^T \times T$ by shifts, i.e.
\[ g \cdot (\mathbf{x}, a) = (\mathbf{x}^g, (g \upharpoonright \ell(a)) \cdot a), \; \text{where} \; \mathbf{x}^g(b) = \mathbf{x}((g \upharpoonright \ell(b))^{-1}b). \]
which is continuous with respect to this topology. We will associate the $G_T$-space $\mathbb{R}^T$ with $(\mathbb{R}^T \times T^0)$.

We will think of elements of $\mathbb{R}^T \times T$ as countable structures in some language, and we will say $(\mathbf{x}, a) \cong (\mathbf{y}, b)$ iff $(\mathbf{y}, b) \in G_T \cdot (\mathbf{x}, a)$. We will always write elements $\mathbf{x} \in \mathbb{R}^T$ in bold.

Given a Polish group $G$ and Polish $G$-space $X$ we define for an arbitrary subset $A \subseteq X$ and nonempty open $V \subseteq G$ the sets $A^{\Delta V}$ and $A^{*V}$, called the {\bf Vaught transforms} of $A$, by:
\[A^{\Delta V} := \{x \in X \mid \text{for a nonmeager set of } g \in V, \; g \cdot x \in A\}\]
and
\[\quad A^{* V} := \{x \in X \mid \text{for a comeager set of } g \in V, \; g \cdot x \in A\} .\]
See \cite[Lemma 5.1.7]{Becker_Kechris_1996} for some basic properties of Vaught transforms. In the case of the space $\mathbb{R}^T$ with the action of $G_T$, we will use the special notation $A^{\Delta a}$ and $A^{* a}$ for subset $A \subseteq \mathbb{R}^T$ and $a \in T$ to refer to the sets $A^{\Delta V^T_{a}}$ and $A^{* V^T_{a}}$ respectively.

\subsection{Inverses of Borel reductions}
Given an equivalence relation, we may want to show that it is Borel-reducible to an action of a Polish group whose orbit equivalence relation has a certain Borel complexity. In this subsection, we state a result that makes this easy.

We will now move onto proving Theorem \ref{th:borel_reduction_inverse}. 

Given a Polish space $X$ and any subset $A \subseteq X$, we say that a set $B \subseteq X$ is \emph{Borel relative to $A$} iff there is a Borel set $B_0 \subseteq X$ such that $B_0 \cap A = B \cap A$. This is the same as saying that $B \cap A$ is Borel in the subspace topology on $A$.

\begin{definition}
Suppose $X$ and $Y$ are Polish spaces and $A \subseteq X$ any subset. A function $\Phi : A \rightarrow \mathcal{P}(\mathcal{P}(Y))$ is \emph{Borel-on-Borel relative to $A$} iff for every Polish space $Z$ and Borel $B \subseteq Z \times X \times Y$, the set $\{(z, x) \in Z \times A \mid B_{z, x} \in \Phi(x)\}$ is Borel relative to $Z \times A$.
\end{definition}

We will need a slightly stronger version of the standard ``large-section" uniformization result (see \cite[Theorem 18.6]{Kechris_1997}) for an analytic domain. The same proof works with only minor modifications, but for completeness, we sketch a proof of the uniformization result that we will need.

\begin{proposition}\label{pr:large_section}
Suppose $P \subseteq X \times Y$ is Borel for Polish spaces $X$ and $Y$, and for every $x$, there is a $\sigma$-ideal $I_x$ on $Y$ such that $P_x \not\in I_x$ whenever $P_x$ is nonempty. Assume also that the map $x \rightarrow I_x$ is Borel-on-Borel relative to $\text{Proj}_X(P)$. Then there is a Borel uniformization of $P$.
\end{proposition}

\begin{proof}
Fix a Borel injection $f : \omega^\omega \rightarrow X \times Y$ such that $f[\omega^\omega] = P$.
For $s \in \omega^{<\omega}$, fix some Borel set $B_s \subseteq X$ such that
\[x \in B_s \quad \text{iff} \quad (f[N_s])_x \not\in I_x\]
for every $x \in \text{Proj}_X(P)$.
Define a Borel map $x \mapsto T_x$ from $X$ to the (Polish) space of trees on $\omega$ such that 
\[s \in T_x \quad \text{iff} \quad x \in B_s\]
for every $s \in \omega^{<\omega}$.
Define $D \subseteq X$ to be the set of $x \in X$ such that $T_x$ codes a nonempty pruned tree. Then $D$ is Borel and $D \supseteq \text{Proj}_X(P)$ and we can define a Borel map $h : D \rightarrow \omega^\omega$ sending $x$ to the left-most path in the tree coded by $T_x$. Then the map from $x$ to the unique $y$ such that $f(h(x)) = (x, y)$ is a Borel map from $D$ to $Y$ which restricts to an Borel uniformization of $P$.
\end{proof}

We recall the following definition.

\begin{definition}
An equivalence relation $E$ on a Polish space $X$ is \emph{idealistic} if there is a $E$-invariant, Borel-on-Borel map $x \mapsto I_{[x]_E}$ such that each $I_{[x]_E}$ is a $\sigma$-ideal on $X$ where $[x]_E \not\in I_{[x]_E}$.
\end{definition}

Note that equivalence relations induced by Borel actions of Polish groups are idealistic (see \cite[5.4.10]{Gao_2009}). 

We now move on to Theorem \ref{th:borel_reduction_inverse}. This theorem is very similar to a theorem that appeared in \cite{Gao_2001}, which is stated as folklore and attributed to arguments of Burgess and Kechris.

\begin{theorem}\label{th:borel_reduction_inverse}
Suppose $E_1$, $E_2$, and $F$ are Borel equivalence relations on Polish spaces $X_1$, $X_2$, and $Y$ respectively, with the additional assumption that $E_2$ is idealistic. Let $f : Y \rightarrow X$ be a Borel reduction of $F$ to $E_1$ and assume also $F \le_B E_2$. Then there is a $E_1$-invariant Borel set $B \subseteq X_1$ such that $f[Y] \subseteq B$ and $E_1 \upharpoonright B \le_B E_2$.
\end{theorem}



\begin{proof}
Fix a Borel reduction $g : Y \rightarrow X_2$ from $F$ to $E_2$, and let $\Phi \subseteq \mathcal{P}(X_1 \times X_2)$ such that 
\[\Phi(P) \quad \text{iff} \quad \forall (x_1, x_2), (x_1', x_2') \in P, \; (x_1 E_1 x_1' \leftrightarrow x_2 E_2 x_2').\] 

Define $P_0 \subseteq X_1 \times X_2$ by
\[P_0 := \{(x_1, x_2) \mid \exists y \in Y, f(y) E_1 x_1 \; \text{and} \; g(y) E_2 x_2\}. \]
Because $\Phi$ is $\mathbf{\Pi}^1_1$-on-$\mathbf{\Sigma}^1_1$ and $\Phi(P_0)$, we can find some Borel $P_1 \supseteq P_0$ such that $\Phi(P_1)$ by the First Reflection Theorem (see \cite[35.10]{Kechris_1997}). 

Fix a Borel-on-Borel map $x_2 \mapsto I_{[x_2]_{E_2}}$ witnessing the fact that $E_2$ is idealistic. For $x_1 \in \text{Proj}_{X_1}(P_1)$, define an ideal $J_{x_1}$ on $Y$ where 
\[A \in J_{x_1}\quad \text{iff} \quad \exists x_2, \; (x_1, x_2) \in P_1 \;  \text{and} \; A \in I_{[x_2]_{E_2}}. \]
Define
\[P := \{(x_1, x_2) \in P_1 \mid (P_1)_{x_1} \not\in J_{x_1}\} = \{(x_1, x_2) \mid (P_1)_{x_1} \not\in I_{[x_2]_{E_2}}\},\]
which is a Borel set.

We claim that the map $x_1 \mapsto J_{x_1}$ is Borel-on-Borel relative to $\text{Proj}_{X_1}(P)$. To see this, fix some Polish space $Z$ and a Borel subset $B \subseteq Z \times X_1 \times X_2$. Let
\[A_1 := \{(z, x_1) \in P \mid \exists x_2 \in X_2, (x_1, x_2) \in P \; \text{and} \; B_{z, x_1} \in I_{[x_2]_{E_2}}\}\]
and
\[A_2 := \{(z, x_1) \in Z \times X_1 \mid \forall x_2 \in X_2, \; ((x_1, x_2) \in P  \rightarrow B_{z, x_1} \in I_{[x_2]_{E_2}})\}.\]
By analytic separation, fix some Borel set $A \supseteq A_1$ such that $A \subseteq A_2 = \emptyset$. Observe that 
\[A_1 \cap (Z \times \text{Proj}_{X_1}(P)) = A_2 \cap (Z \times \text{Proj}_{X_1}(P)) = A \cap (Z \times \text{Proj}_{X_1}(P)) \]
and so $A_1$ is Borel relative to $\text{Proj}_{X_1}(P)$.

By Proposition \ref{pr:large_section}, we can fix some Borel uniformization $h : B \rightarrow X_2$ of $P$, where $B := Proj_{X_1}(P)$ is Borel. By definition of $P$ this means that $h \upharpoonright B$ is a reduction from $E_1 \upharpoonright B$ to $E_2$, as desired.
\end{proof}

\subsection{Changes of topology}

Given a Polish group $G$ and a Polish $G$-space $X$ such that $E^G_X$ has potential Borel complexity $\Gamma$, by definition there is a Polish topology $\sigma$ on $X$ such that $E^G_X$ is $\mathbf{\Sigma}$ in $X \times X$ in the product topology. However, it is often useful to choose $\sigma$ such that the action of $G$ is continuous with respect to $\sigma$. In the case that $G$ is non-Archimedean, this is possible.

The next proposition is essentially \cite[4.3.3]{Gao_2009}, due to Hjorth, but requires a small argument to get the exact statement that we want.


\begin{proposition}\label{pr:hjorth_topology_change}(Hjorth)
Let $\alpha < \omega_1$ and let $\mathcal{A}$ be a countable family of subsets of $X$ such that for each $A \in \mathcal{A}$, there is some $\beta < \alpha$ such that $A$ is $\mathbf{\Sigma}^0_{\beta}$. Then there is a finer topology $\sigma$ such that for every $A \in \mathcal{A}$ and basic open $V \subseteq G$, $A^{\Delta V}$ is open in $\sigma$. Furthermore, every set open in $\sigma$ is $\mathbf{\Sigma}^0_\beta$ for some $\beta < \alpha$ in the original topology.
\end{proposition}

\begin{proof}
When $\alpha$ is a successor ordinal, this follows immediately from \cite[4.3.3]{Gao_2009}. For limit $\alpha$, we do a tiny bit more work.

For each $\beta < \alpha$, let $\mathcal{A}_\beta$ be the elements of $\mathcal{A}$ that are $\mathbf{\Sigma}^0_\beta$.  By \cite[4.3.3]{Gao_2009}, there is a finer topology $\sigma_\beta$ such that each $A^{\Delta V}$ is open for $A \in \mathcal{A}_\beta$ and $V \in \mathcal{B}$, and every open set in $\sigma_\beta$ is $\mathbf{\Sigma}^0_\beta$ in the original topology.

Let $\sigma = \bigcup_{\beta < \alpha} \sigma_\beta$. By \cite[4.3.2]{Gao_2009}, this has the desired properties.
\end{proof}

\begin{lemma}\label{lm:top_change}
Let $G$ be a Polish group, $X$ a Borel $G$-space with Polish topologies $\tau$ and $\sigma$. Assume furthermore that $\mathcal{B}_0$ is a basis of open subsets of $G$ and that $U^{\Delta V}$ is open in $\sigma$ for every $U \subseteq X$ that is open in $\tau$.

Then for any basic open $V \subseteq G$, any ordinal $\alpha \ge 1$, and any set $A \subseteq X$,
\[\text{if } A \text{ is } \mathbf{\Pi}^0_\alpha \text{ in } \tau, \quad \text{then} \quad A^{*V} \text{ is } \mathbf{\Pi}^0_\alpha \text{ in } \sigma,\]
and
\[\text{if } A \text{ is } \mathbf{\Sigma}^0_\alpha \text{ in } \tau, \quad \text{then} \quad A^{\Delta V} \text{ is } \mathbf{\Sigma}^0_\alpha \text{ in } \sigma.\]
\end{lemma}

\begin{proof}
Suppose $U$ is open in $\tau$. Then we can write $U = \bigcup_{n \in \omega} U_n$ where each $U_n$ is basic open in $\tau$, and then
\[ U = U^{\Delta V} = (\bigcup_{n \in \omega} U_n)^{\Delta V} = \bigcup_{n \in \omega} U_n^{\Delta V},\]
which is open in $\sigma$.

Fix $\alpha$ and assume for our induction hypothesis that for any basic open $V$ and any $A \subseteq X$ which is $\mathbf{\Sigma}^0_\alpha$ in $\tau$ for some $\beta < \alpha$, $A$ is $\mathbf{\Sigma}^0_\beta$ in $\sigma$. 

If $A \subseteq X$is $\mathbf{\Pi}^0_\beta$ in $\tau$, then its complement is $\mathbf{\Sigma}^0_\beta$ in $\tau$ and thus $A^{\Delta V}$ is $\mathbf{\Sigma}^0_\beta$ in $\sigma$, and so $A^{* V}$ is $\mathbf{\Pi}^0_\beta$ in $\sigma$.

Now, if $A$ is $\mathbf{\Sigma}^0_\alpha$ in $\tau$, then we can write $A = \bigcup_{n \in \omega} A_n$ where for each $n$, there is some $\beta < \alpha$ such that $A_n$ is $\mathbf{\Pi}^0_\beta$. Then
\[A = A^{\Delta V} = (\bigcup_{n \in \omega} A_n)^{\Delta V} = \bigcup_{n \in \omega} A_n^{\Delta V} = \bigcup_{n \in \omega} \bigcup \{A_n^{*W} \mid W \subseteq V \; \text{and} \; W \in \mathcal{B}_0\}. \]
By the induction hypothesis, if $A_n$ is $\mathbf{\Pi}^0_\beta$ in $\tau$, then $A_n^{*W}$ is $\mathbf{\Pi}^0_\beta$ in $\sigma$, and thus $A^{\Delta V}$ is $\mathbf{\Pi}^0_\alpha$ in $\sigma$. The result follows by induction on $\alpha$.
\end{proof}

This allows us to prove the following.
\begin{theorem}\label{th:top_change}
Let $G$ be a Polish group and $X$ a Polish $G$-space. If $E^G_X$ is potentially $\Gamma$ for some $\Gamma$ in $(*)$, then there is a topology $\sigma$ on $X$ such that $(X, \sigma)$ is a Polish $G$-space and $E^G_X$ is $\Gamma$.
\end{theorem}

\begin{proof}
Fix a countable basis $\mathcal{B}_0$ of the topology on $G$. Let $\tau$ be a Polish topology on $X$ such that $E^G_X$ is $\Gamma$ in $\tau$. By Proposition  \ref{pr:hjorth_topology_change}, we can find a topology $\sigma$ such that $(X, \sigma)$ is a Polish $G$-space, and $U^{\Delta V}$ is open in $\sigma$ for every basic open $V \subseteq G$ and every basic $\tau$-open $U \subseteq X$.

Observe that
\[\{ V_1 \times V_2 \mid V_1, V_2 \in \mathcal{B}_0\}\]
is a countable basis of open cosets of $G \times G$ with the product topology, and futhermore, if $U_1 \times U_2$ is open in $\tau \times \tau$, then by Kuratowski-Ulam,
\[(U_1 \times U_2)^{\Delta (V_1 \times V_2)} = U_1^{\Delta V_1} \times U_2^{\Delta V_2}. \]
Thus the natural product action of $G \times G$ on $X \times X$ and the product topologies $\tau \times \tau$ and $\sigma \times \sigma$ satisfy the conditions of Lemma \ref{lm:top_change}. By noting that $E^G_X = (E^G_X)^{\Delta G \times G} = (E^G_X)^{* G \times G}$, it is easy to see for each $\Gamma$ that it follows from the lemma that $E^G_X$ is $\Gamma$ in $\sigma$.
\end{proof}

We can then conclude the following, which we will use extensively.

\begin{corollary}\label{cor:upper_bound}
Suppose $E$ is an equivalence relation which is Borel-reducible to to an orbit equivalence relation $E^G_X$ for some Polish group $G$ and Polish $G$-space $X$, and also Borel-reducible an orbit equivalence relation $E^H_Y$ for some Polish group $H$ and Polish $H$-space $Y$ with potential complexity $\Gamma$ from (*). Then there is a Polish $G$-space $Z$ such that $E$ is Borel-reducible to $E^G_Z$, and $E^G_Z$ has (true) Borel complexity $\Gamma$.
\end{corollary}


\section{The Hjorth analysis of Polish group actions}

In \cite{Hjorth_2010}, Hjorth outlined a method of extending the Scott analysis of isomorphism of countable structures to arbitrary Polish group actions. That work was continued in \cite{Drucker_2015}. We refer to this as the {\bf Hjorth analysis of Polish group actions}.

For the rest of the section, let $G$ be a Polish group and $X$ a Polish $G$-space.

\begin{definition}(Hjorth)
$(x, U) \le^1 (y, V)$ iff $U \cdot x \subseteq \overline{V \cdot y}$.

For $\alpha \ge 1$, $(x, U) \le^{\alpha+1} (y, V)$ iff for every nonempty open $U' \subseteq U$, there is some nonempty open $V' \subseteq V$ such that $(y, V') \le^\alpha (x, U')$.

Finally, for limit $\lambda$, $(x, U) \le^\lambda (y, V)$ iff for every $1 \le \alpha < \lambda$, $(x, U) \le^\alpha (y, U)$.
\end{definition}

\begin{proposition}\label{pr:hjorth_inv}(Hjorth)
$(x, U) \le^\alpha (y, V)$ iff 
\[\text{for every} \; \mathbf{\Pi}^0_\alpha \; \text{set} \; A, \quad y \in A^{* V} \Rightarrow x \in A^{* U}.\]
\end{proposition}

\begin{proof}
\cite[Lemma 1.9]{Hjorth_2010}
\end{proof}

A theorem similar to the following appears in \cite{HKL_1998}, though it is specific to actions of non-Archimedean Polish groups. We prove a more general result, for arbitrary Polish group actions.

\begin{proposition}\label{pr:isom_inv}
Suppose that $A$ and $B$ are both nonempty $\mathbf{\Pi}^0_{\alpha+2}$ and $G$-invariant. Then the following are all equivalent:
\begin{enumerate}
    \item $A \cap B \neq \emptyset$;
    \item For every $\mathbf{\Pi}^0_\alpha$ set $C$, $A \cap C^{\Delta G} \neq \emptyset$ iff $B \cap C^{\Delta G} \neq \emptyset$;
    \item For every $\mathbf{\Sigma}^0_{\alpha+1}$ set $C$, $A \cap C^{\Delta G} \neq \emptyset$ iff $B \cap C^{\Delta G} \neq \emptyset$;
    \item For every $\mathbf{\Pi}^0_{\alpha+1}$ set $C$, $A \cap C^{* G} \neq \emptyset$ iff $B \cap C^{* G} \neq \emptyset$;
\end{enumerate}
\end{proposition}

\begin{proof}
Statement (1) clearly satisfies the others, and the statements (2), (3), and (4) are easily seen to be equivalent by basic properties of the Vaught transform. We now show that (3) implies (1).

By the assumption on the Borel complexities, we may write 
\[A = \bigcap_{k \in \omega} \bigcup_{\ell \in \omega} A_{k, \ell}\]
and 
\[B = \bigcap_{k \in \omega} \bigcup_{\ell \in \omega} B_{k, \ell}\]
where each $A_{k, \ell}$ and $B_{k \ell}$ are $\mathbf{\Pi}^0_\alpha$. By basic properties of the Vaught transforms, fixing some countable basis $\mathcal{B}$ for the topology on $G$, we may write:
\[A = A^{* G} = \bigcap_{k \in \omega} \left[\bigcup_{\ell \in \omega} A_{k, \ell}\right]^{* G} = \bigcap_{k \in \omega} \bigcap_{V \in \mathcal{B}} \left[\bigcup_{\ell \in \omega} A_{k, \ell}\right]^{\Delta V} = \bigcap_{k \in \omega} \bigcap_{V \in \mathcal{B}} \bigcup_{\ell \in \omega} A_{k, \ell}^{\Delta V}. \]
Similarly, we may write:
\[B = \bigcap_{k \in \omega} \bigcap_{V \in \mathcal{B}} \bigcup_{\ell \in \omega} B_{k, \ell}^{\Delta V}. \]

We seek a new topology $\sigma$ on $X$, in which $(X, \sigma)$ is still a Polish $G$-space, where $A_{k, \ell}^{\Delta V}$ and $B_{k, \ell}^{\Delta V}$ are open for every $k, \ell \in \omega$ and basic open $V$. Under such a topology, $A$ and $B$ will be $\mathbf{\Pi}^0_2$ by the previous paragraph. By Proposition \ref{pr:hjorth_topology_change}, such a topology exists, in which every basic open is $\mathbf{\Sigma}^0_{\alpha+1}$ with respect to the original topology. By the assumption (3), $A$ and $B$ are both dense in $\sigma$, thus they have nonempty intersection.
\end{proof}

\subsection{Actions of non-Archimedean Polish groups}

\begin{definition}
Given an equivalence relation $E$ on a Polish space $X$, the \emph{Friedman-Stanley jump} of $E$ is the equivalence relation $E^+$ on $X^\omega$ such that $(x_i)_{i \in \omega} E^+ (y_i)_{i \in \omega}$ iff for every $i \in \omega$ there exists some $j \in \omega$ such that $x_i E y_j$ and also for every $j \in \omega$ there exists some $i \in \omega$ such that $x_i E y_j$.
\end{definition}

For countable ordinals $\alpha$, the relation $=^{+\alpha}$, the \emph{iterated Friedman-Stanley jump of equality}, is defined recursively where $=$ is equality on the reals, $=^{+(\alpha+1)}$ is the Friedman-Stanley jump of $=^{+\alpha}$ and $=^{+\alpha}$ is the product of $=^{+\beta}$ for $\beta < \alpha$ and limit $\alpha$.

\begin{proposition}\label{pr:hkl}(Hjorth-Kechris-Louveau)
Suppose $G$ is non-Archimedean. Then
\begin{enumerate}
    \item $E^G_X \le_B \; =_{\mathbb{R}}$ iff $E^G_X$ is potentially $\mathbf{\Pi}^0_1$;
    \item For every $n \in \omega$, $E^G_X \le_B =^{+(n+1)}$ iff $E^G_X$ is potentially $\mathbf{\Pi}^0_{n+3}$;
    \item For every limit $\lambda$, $E^G_X \le_B =^{+\lambda}$ iff $E^G_X$ is potentially $\mathbf{\Pi}^0_\lambda$; and
    \item For every limit $\lambda$ and every $n \in \omega$, $E^G_X \le_B =^{+(\lambda+n+1)}$ iff $E^G_X$ is potentially $\mathbf{\Pi}^0_{\lambda+n+2}$.
\end{enumerate}
\end{proposition}

\begin{proof}
\cite[Theorem 2]{HKL_1998}
\end{proof}

\subsection{Actions of TSI Polish groups}

We prove an important fact in the case that $G$ is TSI. We first recall an important lemma from a paper of Hjorth and Solecki.

\begin{proposition}\label{pr:hjorth_solecki}(Hjorth-Solecki) Suppose $G$ is TSI and $x,y \in X$ satisfy $\overline{G\cdot x} = \overline{G \cdot y}$. If $G \cdot x$ is $\mathbf{\Pi}^0_2$ then $G \cdot x = G \cdot y$.
\end{proposition}

\begin{proof}
See \cite[Lemma 3.5]{Hjorth_Solecki_1999}.
\end{proof}

This allows us to improve Proposition \ref{pr:isom_inv}.

\begin{proposition} Suppose that $G$ is TSI. Then the assumption of Proposition \ref{pr:isom_inv} can be weakened such that only $A$ need be $\mathbf{\Pi}^0_{\alpha+2}$.
\end{proposition}

\begin{proof}
The proof of Proposition \ref{pr:isom_inv} used the fact that two $\mathbf{\Pi}^0_2$ sets with the same closure must intersect. Simply replace that fact with an application of Proposition \ref{pr:hjorth_solecki}.
\end{proof}






\section{Universal equivalence relations}
Given a Polish group $G$ that is both non-Archimedean and TSI, we would like to define a Polish $G$-space whose orbit equivalence relation is universal for Borel actions of $G$, along with a family of coarsenings that are universal for orbit equivalence relations induced by Borel actions of $G$ for each potential complexity class.

Fixing a group tree $T$ such that $G_T$ is homeomorphic to $G$, our goal in this section is then to prove that $\cong$ on $\mathbb{R}^T$ for a given group tree $T$ is universal for all Borel actions of $G^T$ Polish groups.

\begin{theorem}\label{th:universal_map}
Let $X$ be a Polish $G_T$-space. Then there is a Borel injective map $f : X \rightarrow \mathbb{R}^T$ which is equivariant, i.e.
\[ \forall g \in G_T, \forall x \in X, g \cdot f(x) = f(g \cdot x). \]
\end{theorem}

\begin{proof}
Since there is a continuous injection of Cantor space onto a closed subset of $\mathbb{R}$, it is enough to construct a map $f : X \rightarrow (2^\omega)^T$, which satisfies all of the required properties.

Let $\mathcal{A}$ be any countable family of Borel subsets of $X$ containing the basic open sets and closed under finite unions and intersections, with enumeration $\mathcal{A} = \{A_n \mid n \in \omega\}$. Let $f : X \rightarrow (2^\omega)^T$ be the function defined by 
\[f(x) = \mathbf{x} \text{ where } \mathbf{x}(a)(n) = 1 \text{ iff } x \in A_n^{* a}\]
We claim that $f$ works. Injectivity follows from the fact that $X$ is Hausdorff and the action is continuous.

To see that $f$ is Borel, fix an arbitrary basic open $U_c \subseteq (2^\omega)^T$ where
\[ U_c = \{y \in (2^\omega)^T \mid \forall a \in dom(f), x(a) \sqsupseteq c(a)\} \]
for some finite partial function $c$ from $T$ to $2^{<\omega}$. Then
\begin{equation}\label{eq:1}
f^{-1}[U_c] = \bigcap \big[\{A^{*a}_n \mid c(a)(n) = 1\} \cup \{X \setminus A^{* a}_n \mid c(a)(n) = 0\}\big],
\end{equation}
which is Borel.

To show equivariance, fix arbitrary $g \in G_T$, $x \in X$, $n \in \omega$, and $a \in T$ of length $k$, and observe that
\begin{align*}
    &[g \cdot f(x)](a)(n) = 1 \\
    \iff &x \in A_n^{*[(g \upharpoonright k)^{-1} a]} \\
    \iff &(\forall^* h \in a^{-1} (g \upharpoonright k) V^T_n) \; hx \in A_n\\
    \iff &(\forall^* h \in a^{-1} V^T_n g) \; hx \in A_n \\
    \iff &gx \in A_n^{*a} \\
    \iff &f(g \cdot x)(a)(n) = 1
\end{align*}
as desired, using the fact that $V^T_k$ is a normal subgroup.
\end{proof}

We will now define, for a given group tree $T$, relations $\sim^T_\alpha$ on $\mathbb{R}^T$ for $\alpha < \omega_1$. 
We will define them recursively on $\alpha$, and the recursive definitions will actually require us to define them on the slightly larger space $\mathbb{R}^T \times T$. 
We may think about this space as the space of countable structures in a certain language, with a certain distinguished tuple of elements. 
In this sense, we may view the relation $\sim^T_\alpha$ as asserting $\alpha$-back-and-forth equivalence in the sense of the Scott Analysis (see \cite[Chapter 12]{Gao_2009}).
We will also need to define ``relativized" relations $\sim^T_{\alpha, n}$ for $n \in \omega$, where $\sim^T_\alpha$ is identified with $\sim^T_{\alpha, 0}$. In fact, we will want to define $\sim^{T, \sigma}_{\alpha, 0}$ for different topologies $\sigma$ on $\mathbb{R}^T$. In the case that $\sigma$ is the usual topology on $\mathbb{R}^T$, we will suppress it.

\begin{definition}
By transfinite recursion, define relations $\sim^{T, \sigma}_{\alpha, n}$ on $\mathbb{R}^T \times T$ for ordinals $\alpha$ and $n \in \omega$ by:
\[(\mathbf{x}, a) \sim^{T, \sigma}_0 (\mathbf{y}, b)\]
iff
\[ \overline{V^T_a \cdot \mathbf{x}} = \overline{V^T_b \cdot \mathbf{y}} \; \text{and} \; \ell(a) = \ell(b) \]
and for any $\alpha \ge 1$,
\[(\mathbf{x}, a) \sim^{T, \sigma}_{\alpha, n} (\mathbf{y}, b)\]
iff
\[\exists c \in T^n_a, \exists d \in T^n_b, \forall \beta < \alpha, \forall m \in \omega, \; (\mathbf{x}, a \smallfrown c) \sim^T_{\beta, m} (\mathbf{y}, b \smallfrown d). \]
\end{definition}

The following are easily verified:
\begin{lemma}\label{sim_facts}
For any ordinal $\alpha$ and $n \in \omega$:
\begin{enumerate}
    \item If $(\mathbf{x}, a) \sim^T_{\alpha, n} (\mathbf{y}, b)$ then $(\mathbf{x}, a \cdot c) \sim^T_{\alpha, n} (\mathbf{y}, b \cdot c)$
    \item $(\mathbf{x}, a) \sim^T_{\alpha, 0} (\mathbf{y}, b)$ iff $(\mathbf{x}, a) \sim^T_\alpha (\mathbf{y}, b)$; and
    \item If $(\mathbf{x}, a) \sim^T_{\alpha, n} (\mathbf{y}, b)$ then $(\mathbf{x}, a) \sim^T_{\beta, m} (\mathbf{y}, b)$ for any $\beta < \alpha$ and any $m \in \omega$;
    \item If $(\mathbf{x}, a) \sim^T_{\alpha, n} (\mathbf{y}, b)$ then $(\mathbf{x}, a) \sim^T_{\alpha, m} (\mathbf{y}, b)$ for any $m < n$;
\end{enumerate}
\end{lemma}

\begin{proposition}\label{prop:friedman_stanley}For any $\alpha < \omega_1$, $\sim^T_\alpha$ is Borel reducible to $=^{+(\alpha)}$, i.e. the $\alpha$th Friedman-Stanley jump of equality.
\end{proposition}

\begin{proof}
We will prove this by describing how, for each ordinal $\alpha$, to associate each $(\mathbf{x}, a) \in \mathbb{R}^T \times T$ with a set $A^\alpha_{\mathbf{x}, a} \in \mathcal{P}^\alpha(\mathbb{R})$ such that $(\mathbf{x}, a) \sim^T_\alpha (\mathbf{y}, b)$ iff $A^\alpha_{\mathbf{x}, a} = A^\alpha_{\mathbf{y}, b}$.

For any $(\mathbf{x}, a) \in \mathbb{R}^T \times T$, we can represent $(\mathbf{x}, a)$ by the subset 
\[A^0_{x, a} := \{f \in \mathcal{F} \mid \exists g \in V^T_n, a^{-1}g \cdot x \in U_f\}.\]
By fixing an enumeration of $\mathcal{F}$, this defines a Borel reduction of $\sim^T_0$ to equality on the reals.

Now suppose for some $\alpha$ that the reduction has been defined for $\beta$ for every $\beta < \alpha$. For every $(\mathbf{x}, a) \in \mathbb{R}^T \times T$ and $\beta < \alpha$, let $A^\beta_{(\mathbf{x}, a)}$ be the element of $\mathcal{P}^\alpha(\mathbb{R})$ that serves as an invariant for its $\sim^T_\beta$-class. Then for any $(\mathbf{x}, a)$, the invariant for its $\sim^T_{\alpha}$ class is simply the set
\[ A^{\alpha} :=  \{A^\beta_{(\mathbf{x}, a \smallfrown b)} \mid b \in T_a \; \text{and} \; \beta < \alpha \}. \]
The fact that the reduction is clear from the definition of $\sim^T_{\alpha+1}$. The limit case is immediate.
\end{proof}

\begin{lemma}\label{lm:hjorth_isom}
Fix points $(\mathbf{x}, a), (\mathbf{y}, b) \in \mathbb{R}^T \times T$ with $\ell(a) = m$, $\ell(b) = n$, and $m \ge n$. Then, interpreting the Hjorth relations $\le^\alpha$ as computed in the space $(\mathbb{R}^T, \sigma)$, we have
\[ (\mathbf{x}, a\upharpoonright n) \sim^{T, \sigma}_{k} (\mathbf{y}, b) \; \text{iff} \; (\mathbf{x}, a V^T_m) \le^{k+1} (\mathbf{y}, b V^T_n) \]
for $k \in \omega$, and
\[ (\mathbf{x}, a\upharpoonright n) \sim^{T, \sigma}_{\lambda+ k} (\mathbf{y}, b) \; \text{iff} \; (\mathbf{x}, a V^T_m) \le^{\lambda + k+1} (\mathbf{y}, b V^T_n) \]
for limit $\lambda < \omega_1$ and $k \in \omega$.
\end{lemma}

\begin{proof}
First, we prove the base case. Suppose that 
\[(\mathbf{x}, a \upharpoonright n) \sim^T_1 (\mathbf{y}, b).\]
We want to show that 
\[(\mathbf{x}, a V^T_m) \le^2 (\mathbf{y}, b V^T_n),\]
so fix some nonempty open $W \subseteq a V^T_m$. 

For large enough $k$, there is some $c \in T^m_a$ such that $(a \smallfrown c) V^T_{m+k} \subseteq W$. Then we can find some $d \in T^n_b$ of length $\ell$ such that $(\mathbf{x}, a \smallfrown c) \sim^T_0 (\mathbf{y}, b \smallfrown d)$. 
Then we have 
\[\overline{(b \smallfrown d) V^T_{m+k} \mathbf{y}} = \overline{(a \smallfrown c) V^T_{n+\ell} \mathbf{x}} \subseteq \overline{W x}\]
as desired.

Now suppose that 
\[(\mathbf{x}, a V^T_m) \le^2 (\mathbf{y}, b V^T_n).\]
For any $c \in T_a$ of length $k$, we can find some nonempty open $W \subseteq b V^T_n$ such that 
\[(\mathbf{y}, W) \le^1 (\mathbf{x}, (a \smallfrown c) V^T_{m+k}).\]
For some large enough $\ell$, we can find some $d \in T_{b}^\ell$ such that $(b \smallfrown d)V^T_{n+\ell} \subseteq W$, in which case we have 
\[\overline{(b \smallfrown d)V^T_{n+\ell}\mathbf{y}} \subseteq \overline{(a \smallfrown c)V^T_{m+k}\mathbf{x}}.\]
In particular, assuming without loss of generality that $n+\ell \ge m+k$, we have 
\[\mathbf{y}((b \smallfrown d )\upharpoonright m+ k)) = \mathbf{x}(a \smallfrown c)\]
and thus 
\[(\mathbf{x}, a \smallfrown c) \sim^T_0 (\mathbf{y}, b \smallfrown (d \upharpoonright m+k-\ell))\]
as desired.

Now we proceed to the induction step. Suppose for some ordinal $\alpha$ the claim is true for every $\beta < \alpha$. Assume first that 
\[(\mathbf{x}, a \upharpoonright n) \sim^T_\alpha (\mathbf{y}, b).\]
We will show that 
\[(\mathbf{x}, a V^T_m) \le (\mathbf{y}, b V^T_n).\]
Fix an arbitrary nonempty open subset $W$ of $a V^T_m$. As before, we will assume without loss of generality that $W = (a \smallfrown c) V^T_{m+k}$ for some $k \in \omega$ and $c \in T_a^k$. By assumption, for an arbitrary ordinal $\beta < \alpha$, we can find some $d \in T_b$ such that 
\[(\mathbf{x}, a \smallfrown c) \sim^T_\beta (\mathbf{y}, b \smallfrown d),\]
and thus 
\[(\mathbf{y}, (b \smallfrown d) V^T_{m+k}) \le^\beta (\mathbf{x}, (a \smallfrown c) V^T_{m+k})\]
by the induction hypothesis. We're done.

Assume finally that 
\[(\mathbf{x}, a V^T_m) \le^\alpha (\mathbf{y}, b V^T_n).\]
Fix some $c \in T_a$ of length $k$ and some $\beta < \alpha$, in which case as before we can find some $d \in T_b$ of length $\ell$ such that 
\[(\mathbf{y}, (b \smallfrown d) V^T_{n+\ell}) \le^\beta (\mathbf{x}, (a \smallfrown c) V^T_{m+k}).\]
Without loss of generality, we may assume $n+\ell \ge m+k$, in which we can apply the induction hypothesis to get 
\[(\mathbf{y}, (b \smallfrown d) \upharpoonright m+k) \sim^T_\beta (\mathbf{x}, a \smallfrown c)\]
and thus 
\[(\mathbf{x}, a \smallfrown c) \sim^T_\alpha (\mathbf{y}, b \smallfrown (d \upharpoonright m+k-\ell)).\]
\end{proof}

\begin{proposition}\label{pr:isom_sim}
Let $A, B \subseteq \mathbb{R}^T \times T$ be any two $\cong$-invariant subsets, $\lambda < \omega_1$ a limit ordinal and $n \in \omega$.
\begin{enumerate}
    \item Suppose $A$ is $\mathbf{\Pi}^0_{n+2}$ in $\sigma$. If there exists $(\mathbf{x}, a) \in A$ and $(\mathbf{y}, b) \in B$ such that $(\mathbf{x}, a) \sim^{T, \sigma}_n (\mathbf{y}, b)$, then $A \cap B \neq \emptyset$;
    \item Suppose $A$ is $\mathbf{\Pi}^0_{\lambda+n+1}$ in $\sigma$. If there exists $(\mathbf{x}, a) \in A$ and $(\mathbf{y}, b) \in B$ such that $(\mathbf{x}, a) \sim^{T, \sigma}_{\lambda+n} (\mathbf{y}, b)$, then $A \cap B \neq \emptyset$;
\end{enumerate}
\end{proposition}

\begin{proof}
This is just Proposition \ref{pr:isom_inv}, Proposition \ref{pr:hjorth_inv}, and Lemma \ref{lm:hjorth_isom}.
\end{proof}

\begin{proposition}\label{pr:sim_reduce}
Let $\sigma$ be any finer topology on $\mathbb{R}^T$ such that $(\mathbb{R}^T, \sigma)$ is a Polish $G_T$-space. Then for any $\alpha$ and $n \in \omega$,
\[ \sim^{T, \sigma}_\alpha \le_B \sim^T_\alpha\]
and
\[ \sim^{T, \sigma}_{\alpha, n} \le_B \sim^T_{\alpha, n}. \]
\end{proposition}

\begin{proof}
Let $f : \mathbb{R}^T \rightarrow \mathbb{R}^T$ be the Borel reduction from Theorem \ref{th:universal_map}. It's routine to check that $f$ witnesses all of the desired Borel reductions.
\end{proof}

Given a sequence $(E_n)_n$ of equivalence relations living on Polish spaces $X_n$, let $\bigsqcup_n E_n$, the \emph{disjoint union} of the sequence $(E_n)_n$ be the equivalence relation living on the disjoint union $\bigsqcup_n X_n$ that relates $x$ and $y$ iff for some $n$, $x, y \in X_n$ and $x E_n y$. Then we define
\[ \sim^T_{\alpha, <\omega} = \bigsqcup_n \sim^T_{\alpha, n}. \]

We say that for a Polish group $G$ and a Borel complexity class $\Gamma$ an equivalence relation is \emph{universal for $\Gamma$ actions of $G$} iff $E$ is Borel-reducible to a potentially $\Gamma$ orbit equivalence relation induced by a continuous action of $G$, and furthermore, any such orbit equivalence relation Borel reduces to $E$.

\begin{proposition}\label{pr:fine_universal}
For any group tree $T$ the following all hold:
\begin{enumerate}
    \item For each limit $\lambda < \omega_1$ and $n \in \omega$,
    \begin{enumerate}
        \item $\sim^T_{n}$ is universal for $\mathbf{\Pi}^0_{n+2}$ actions of $G_T^\omega$;
        \item $\sim^T_{\lambda+n}$ is universal for $\mathbf{\Pi}^0_{\lambda+n+1}$ actions of $G_T^\omega$;
    \end{enumerate}
    
    \item For each limit $\lambda < \omega_1$ and $n \in \omega$,
    \begin{enumerate}
        \item $\sim^T_{n, <\omega}$ is universal for $\mathbf{\Sigma}^0_{n+3}$ actions of $G_T^\omega$;
        \item $\sim^T_{\lambda + n, <\omega}$ is universal for $\mathbf{\Sigma}^0_{\lambda+n+2}$ actions of $G_T^\omega$;
    \end{enumerate}
    \item For each limit $\lambda$, $\sim^T_{<\lambda}$ is universal for $\bigoplus_{\alpha < \lambda} \mathbf{\Pi}^0_\alpha$ actions of $G_T^\omega$;
\end{enumerate}
\end{proposition}

\begin{proof}
In the next section, we will show that all of the above relations are Borel-reducible to an an orbit equivalence relation induced by $G_T^\omega$. In Proposition \ref{prop:friedman_stanley}, we showed that $\sim^T_\alpha$ is Borel-reducible to $=^{+\alpha}$ for every $\alpha$. By the calculations from \cite{HKL_1998}[Theorem 5.1] of the Friedman-Stanley jumps of equality, we can compute the complexity upper-bounds. And then an application of Corollary \ref{cor:upper_bound} tells us that the above relations are Borel-reducible to actions of $G_T^\omega$ with the desired complexity.

We move on to the lower-bound calculations. First, we prove part (1). Let $X$ be a Polish $G_T$-space and $n \in \omega$ such that $E^{G_T}_X$ is $\mathbf{\Pi}^0_{n+2}$. Let $f : X \rightarrow \mathbb{R}^T$ be a Borel reduction $E^G_X$ to $\cong$, which exists by Theorem  \ref{th:universal_map}. By Theorem \ref{th:borel_reduction_inverse}, there is a $\cong$-invariant Borel set $B \subseteq \mathbb{R}^T$ containing $f[X]$ such that $\cong \upharpoonright B$ is potentially $\mathbf{\Pi}^0_{n+2}$. By Theorem \ref{th:top_change}, there is a finer topology $\sigma$ such that $(X, \sigma)$ is a Polish $G_T$-space and $\cong \upharpoonright B$ is $\mathbf{\Pi}^0_{n+2}$ in $\sigma$. Then on $B$, $\sim^{T, \sigma}_n$ and $\cong$ coincide, by Proposition \ref{pr:isom_sim}. Thus $E^G_X$ is Borel-reducible to $\sim^{T, \sigma}_n$, which is Borel-reducible to $\sim^T_n$ by Proposition \ref{pr:sim_reduce}. We just proved (1)(a). Part (1)(b) is essentially the same.

We now proceed to part (2). Suppose $X$ is a Polish $G_T$-space such that $E^{G_T}_X$ is $\mathbf{\Sigma}^0_{n+3}$. 
As before, let $f : X \rightarrow \mathbb{R}^T$ be a Borel reduction from $E^{G_T}_X$ to $\cong$ and $B \subseteq \mathbb{R}^T$ a $\cong$-invariant Borel set and $\sigma$ a topology such that $(\mathbb{R}^T, \sigma)$ a Polish $G_T$-space such that $\cong \upharpoonright B$ is $\mathbf{\Sigma}^0_{n+3}$ in $\sigma$. 

Write $\cong \upharpoonright B = \bigcup_\ell A_\ell$ where each $A_\ell$ is $\mathbf{\Sigma}^0_{n+2}$ in $\sigma$. 
For any $x \in B$, we have $(x, x) \in \cong^{\Delta G_T \times G_T}$ where we consider the natural product action of $G_T \times G_T$ on $\mathbb{R}^T \times \mathbb{R}^T$. Thus for some $a, b \in T$ and $\ell$, we have $(x, x) \in A_\ell^{* a, b}$, where we use $A^{*a, b}$ as shorthand notation for 
\[A^{*V^T_a \times V^T_b}.\]
We can assume without loss of generality that $a$ and $b$ have the same length, so that by the continuity of the group action, we have 
\[(x, x) \in A_{\ell}^{* 1^k, 1^k}\]
and thus
\[[(x, 1^k)]_{\cong} = V^T_{1^k} x = [A^{*1^k, 1^k}]_x,\]
which is a $\mathbf{\Pi}^0_{n+2}$ set in $\sigma$.

Therefore, if we define
\[A_k := \{x \in \mathbb{R}^T \mid \exists \ell, (x, x) \in A_\ell^{*1^k, 1^k}\}\]
then each $A_k$ is $G_T$-invariant and $\cong$ and $\sim^{T, \sigma}_{n+2, k}$ coincide on $A_k$. Thus, as before, $E^{G_T}_X \le_B \sim_{n, < \omega}$. This covers 2(a), and 2(b) follows by the same argument.

For part (3), fix a partition $X = \bigsqcup_{\alpha < \lambda} X_\alpha$ of $\cong$-invariant Borel sets such that for every $x \in X_\alpha$, $G \cdot x$ is $\mathbf{\Pi}^0_\alpha$. On $X_\alpha$, the relations $\sim^T_\alpha$ and $\cong$ coincide, thus $E^G_X \le_B \sim_{<\lambda}$.
\end{proof}













\section{A family of actions of non-Archimedean tsi groups}

For this section, fix a group tree $T$. For each $k < m$, we will define $N^m_k$ to be the normal subgroup of $T^m$ defined by:
\[N^m_k := \{a \in T^m \mid a \upharpoonright k = 1^k\}.\]

For a countable group $\Gamma$, the $\Gamma$-shift of a Polish $G$-space $(X, a)$, is the Polish $(\Gamma \times G)$-space $X^T$ with the action defined by:
\[((a, g)\cdot x)(b) = g \cdot x(a^{-1}b). \]

If for each $n$, there is a Polish group $G_n$ and a Polish $G_n$-space $(X_n, a_n)$, write $\prod_n (X_n, a_n)$ to refer to be the Polish $\prod_n G_n$-space living on $\prod_n X_n$ with the induced product action. Write $\bigsqcup_n (X_n, a_n)$ to be the Polish $\prod_n G_n$-space living on $\bigsqcup_n X_n$ with the action induced by restrictions.

We will define a continuous action $a^m_{\alpha, n}$ of a TSI Polish group $G^m_{\alpha, n}$ on a Polish space $X^m_{\alpha, n}$, for each ordinal $\alpha \in \omega_1$ and $n, m \in \omega$.

\begin{definition} For $n, m \in \omega$ and $\alpha \ge 1$, we define the following:
\begin{enumerate}
    \item Let $(X^m_{0, n}, a^m_{0, n})$ be the $T^{m+n} / N^{m+n}_m$-shift of the trivial group acting on $\mathbb{R}^{T^{m+n}}$. This is a Polish $G^m_{0, n}$-space, where $G^m_{0, n}$ is $T^{m+n} / T^m$;
    \item Let $(X^m_{\alpha, n}, a^m_{\alpha, n})$ be the $(T^{m+n}/T^m)$-shift of 
    \[\prod_{i \ge n} \prod_{\beta < \alpha} (X^{m+n}_{\beta, i}, a^{m+n}_{\beta, i}).\]
    This is a Polish $G^m_{\alpha, n}$-space, where $G^m_{\alpha, n}$ is 
    \[(T^{m+n}/T^m) \times \prod_{i \ge n} \prod_{\beta < \alpha} G^{m+n}_{\beta, i}\Big.\]
\end{enumerate}
\end{definition}

For each ordinal $\alpha$, let $E_\alpha$ be the orbit equivalence relation of the $G^0_{\alpha, 0}$-space $(X^0_{\alpha, 0}, a^0_{\alpha, 0})$. Let $E_{\alpha, <\omega}$ be the orbit equivalence relation of $\bigsqcup_n (X^0_{\alpha, n}, a^0_{\alpha, n})$. For limit $\lambda$, let $E_{<\lambda}$ be the orbit equivalence relation of $\bigsqcup_{\beta < \alpha} (X^0_{\beta, 0})$.




\begin{lemma}\label{lm:sim_reduce}
For each ordinal $\alpha$ and $n, m \in \omega$, $\sim^T_{\alpha, n} \upharpoonright (\mathbb{R}^T \times T^m)$ is Borel-reducible to $E^m_{\alpha, n}$.
\end{lemma}

\begin{proof}
First we start with $\alpha = 0$. For $n, m \in \omega$, the reduction $f^0_{n, m} : \mathbb{R}^T \times T^m \rightarrow X^m_{0, n}$ sends $(\mathbf{x}, a) \in \mathbb{R}^T \times T^m$ to $x^a$ where $x^a((1^m \smallfrown b) + T^m) = \mathbf{x}(a \smallfrown b)$.

Now, given $\alpha > 0$, $n, m \in \omega$, the reduction $f^\alpha_{n, m} : \mathbb{R}^T \times T^m \rightarrow X^m_{\alpha, n}$ sends $(\mathbf{x}, a) \in \mathbb{R}^T \times T^m$ to $x^a$ where $x^a(b)(i)(\beta) = f^\beta_{i, n+m}(a \smallfrown b)$.
It is routine to check that these work.
\end{proof}

\begin{proposition}\label{prop:tsi_actions}
For limit $\lambda$ and $n \in \omega$, the relations $\sim^T_{\lambda+n}$ and $\sim^T_{\lambda + n, <\omega}$ are Borel-reducible to actions of $G_T^\omega$.
\end{proposition}

\begin{proof}
It is enough to observe that $G^m_{\alpha, n}$ is isomorphic to $G_T^\omega$ for $\alpha \ge 1$ and isomorphic to $T^n$ for $\alpha = 0$.
\end{proof}


\section{Applications}

We now move on to proving the theorems and corollaries promised in the introduction. Our first goal will be to prove the following.

\tsicomplexityspectrumthm

\begin{proof}
We argue that if a complexity class $\Gamma$ in $(*)$ is not attained, then none of the ones after it are attained. Here, we say that a complexity class $\Gamma$ in $(*)$ is not attained if any orbit eqivalence relation induced by a continuous action of $G$ on a Polish space that has potential complexity $\Gamma$ also has potential complexity $\Gamma'$ for some $\Gamma'$ that appears earlier in $(*)$.

\begin{claim}\label{cl:D_not_attained}
Suppose $D(\mathbf{\Pi}^0_n)$ is not attained for some $n \ge 3$. Then $\sim_\alpha$ is potentially $\mathbf{\Pi}^0_n$ for every $\alpha < \omega_1$.
\end{claim}

\begin{proof}
If $D(\mathbf{\Pi}^0_n)$ is not attained, then $\sim^T_{n-2, k}$ is potentially $\mathbf{\Pi}^0_n$ for every $k$. Thus by its definition, we compute that $\sim^T_{n-1}$ is also potentially $\mathbf{\Pi}^0_n$. In general, for any $\alpha$, if $\sim^T_{\beta, k}$ is potentially $\mathbf{\Pi}^0_n$ for every $\beta < \alpha$ and $k \in \omega$, then so is $\sim^T_\alpha$. By induction, the claim follows.
\end{proof}

\begin{claim}\label{cl:Pi_not_attained}
Suppose $\mathbf{\Pi}^0_n$ is not attained for some $n \ge 3$. Then for every $\alpha < \omega_1$, $\sim_\alpha$ is potentially $\mathbf{\Sigma}^0_{n}$, and in particular, if $n \ge 4$, then for any $\alpha < \omega_1$, $\sim^T_\alpha$ is potentially $D(\mathbf{\Pi}^0_{n-1})$.
\end{claim}

\begin{proof}
If $\mathbf{\Pi}^0_n$ is not attained, then $\sim^T_{n-2}$, which is potentially $\mathbf{\Pi}^0_n$, must be potentially $D(\mathbf{\Pi}^0_{n-1})$. By its definition, $\sim^T_{n-2, k}$ is potentially $\mathbf{\Sigma}^0_n$ and thus potentially $D(\mathbf{\Pi}^0_{n-1})$ for every $k$. In general, for any $\alpha$, if $\sim^T_{\beta, k}$ is potentially $D(\mathbf{\Pi}^0_{n-1})$ for every $\beta < \alpha$ and $k \in \omega$, then $\sim^T_\alpha$ is potentially $\mathbf{\Pi}^0_n$ and thus by the assumption must be potentially $D(\mathbf{\Pi}^0_{n-1})$.
\end{proof}


\begin{claim}
Suppose $\bigoplus_{\alpha < \lambda}\mathbf{\Pi}^0_\alpha$ is not attained for some limit $\lambda$. Then there is some $\alpha < \lambda$ such that for every $\beta < \omega_1$, $\sim_\beta$ is potentially $\mathbf{\Pi}^0_\alpha$.
\end{claim}

\begin{proof}
If $\bigoplus_{\alpha < \lambda}\mathbf{\Pi}^0_\alpha$ is not attained, then there must be some $\alpha < \lambda$ such that $E_{<\lambda}$ is potentially $\mathbf{\Pi}^0_\alpha$. In particular, for every $\beta < \lambda$, we have $\sim^T_\beta$ is potentially $\mathbf{\Pi}^0_\alpha$. Thus, so is $\sim_\lambda$. In general, for any $\gamma$, if $\sim^T_\beta$ is potentially $\mathbf{\Pi}^0_\alpha$ for every $\beta < \gamma$, then $\sim^T_\gamma$ is potentially $\mathbf{\Pi}^0_{\alpha+1}$ by its definition. But $\alpha+ 1 < \lambda$ and so $\sim^T_\gamma$ is potentially $\mathbf{\Pi}^0_\alpha$ as well.
\end{proof}

\begin{claim}
Suppose $\mathbf{\Pi}^0_\lambda$ is not attained for some limit $\lambda$. Then $\sim^T_\alpha$ is potentially $\bigoplus_{\beta < \lambda} \mathbf{\Pi}^0_\beta$ for every $\alpha < \omega_1$.
\end{claim}

\begin{proof}
If $\mathbf{\Pi}^0_\lambda$ is not attained, then $\sim^T_\lambda$ is potentially $\bigoplus_{\alpha < \lambda}\mathbf{\Pi}^0_\alpha$. We argue by induction that $\sim^T_\alpha$ is potentially $\mathbf{\Pi}^0_\lambda$ for every $\alpha < \omega_1$. The claim will then follow.

Fix some $\alpha \ge \lambda$ such that $\sim^T_\alpha$ is potentially $\mathbf{\Pi}^0_\lambda$. Then by assumption there is a Borel partition $\mathbb{R}^T = \bigsqcup_{\beta < \lambda} X_\beta$ into invariant sets such that on $X_\beta$, $\sim^T_\alpha$ is potentially $\mathbf{\Pi}^0_\beta$. Then observe that on $X_\beta$, $\sim^T_{\alpha+1}$ is potentially $\mathbf{\Pi}^0_{\beta+1}$, and thus potentially $\mathbf{\Pi}^0_\lambda$. Since $\sim^T_{\alpha+1}$ is potentially $\mathbf{\Pi}^0_\lambda$ on each $X_\beta$, we have that $\sim^T_{\alpha+1}$ as a whole is potentially $\mathbf{\Pi}^0_\lambda$.

For the limit case $\lambda'$, simply observe that if $\sim_\beta$ is potentially $\mathbf{\Pi}^0_\lambda$ for every $\beta < \lambda'$, then by definition $\sim^T_{\lambda'}$ is too.
\end{proof}

\begin{claim}
Suppose $\mathbf{\Sigma}^0_{\lambda+1}$ is not attained for some limit $\lambda$. Then $\sim^T_\alpha$ is potentially $\mathbf{\Pi}^0_\lambda$ for every $\alpha < \omega_1$.
\end{claim}

\begin{proof}
This is essentially the same as Claim \ref{cl:D_not_attained}.
\end{proof}

\begin{claim}
Suppose $D(\mathbf{\Pi}^0_{\lambda+n})$ is not attained for some limit $\lambda$ and finite $n \ge 2$. Then $\sim^T_\alpha$ is potentially $\mathbf{\Pi}^0_{\lambda+n}$ for every $\alpha < \omega_1$.
\end{claim}

\begin{proof}
This is essentially the same as Claim \ref{cl:D_not_attained}.
\end{proof}

\begin{claim}
Suppose $\mathbf{\Pi}^0_{\lambda+n}$ is not attained for some limit $\lambda$ and finite $n \ge 2$. Then for every $\alpha < \omega_1$, $\sim^T_\alpha$ is potentially $\mathbf{\Sigma}^0_{\lambda+n-1}$, and in particular, for $n \ge 3$, $\sim^T_\alpha$ is potentially $D(\mathbf{\Pi}^0_{\lambda+n})$.
\end{claim}

\begin{proof}
This is essentially the same as Claim \ref{cl:Pi_not_attained}.
\end{proof}

The theorem follows from the above claims, using throughout the result from \cite{HKL_1998} that the list $(*)$ of potential complexity classes is exhaustive for Borel orbit equivalence relations induced by Borel actions of non-Archimedean Polish groups.
\end{proof}

We can now conclude Corollary \ref{cor:tsi_complexity_spectrum}. First, we recall that a Polish group $G$ {\bf involves} a Polish group $H$ if there is a closed subgroup $H'$ of $G$ an a continuous surjective homomorphism from $H'$ onto $H$.

\tsicomplexityspectrumcor

\begin{proof}
By \cite{Solecki_1995}, the groups $\mathbb{Z}^\omega$ and $\mathbb{Z}_p^\omega$ are not tame for any prime $p$. By \cite[Theorem 5.11]{Hjorth_1998}, since these groups are abelian, they have Borel orbit equivalence relations of unbounded complexity in $(*)$. Since $(F_2)^\omega$ involves these groups, we know by \cite[Theorem 2.3.5]{Becker_Kechris_1996} that every orbit equivalence relation induced by a continuous action of either of these groups on a Polish space is Borel bi-reducible with such an action of $(F_2)^\omega$, and so $(F_2)^\omega$ has full Borel potential complexity spectrum as well. Then the previous theorem implies that these groups have full Borel complexity spectrum.
\end{proof}

We now move on to the theorems relating to universality.

\tsiuniversalactionsthm

\begin{proof}
This follows from Proposition \ref{pr:fine_universal} and Theorem \ref{th:borel_reduction_inverse}.
\end{proof}

We will also see now why Corollary \ref{cor:tsi_universal_actions} follows. 

\tsiuniversalactionscor

\begin{proof}

Let $G$ be a non-Archimedean TSI Polish group, and let $T$ be a group tree on $(\Delta_i)_{i \in \omega}$ such that $G_T$ is homeomorphic to $G$. 

Since every countable group is a quotient of $F_\omega$, there is a surjective homomorphism from $(F_\omega)^\omega$ onto $\prod_i \Delta_i$. Since $G_T$ is a closed subgroup of $\prod_i \Delta_i$, we get that $(F_\omega)^\omega$ involves $G_T$.

Similarly, we get that the group $(\mathbb{Q}^{<\omega})^\omega$ involves every non-Archimedean abelian group (for this, see \cite[Corollary 2.5]{Gao_Xuan_2014}). We can then apply Mackey's theorem, \cite[Theorem 2.3.5]{Becker_Kechris_1996}, which implies that any Borel $H$-space is Borel-bireducible with some Borel $G$-space, whenever $G$ involves $H$. Thus if we apply Theorem \ref{th:tsi_universal_actions} with the group $(F_\omega)^\omega$ (resp. $(\mathbb{Q}^{<\omega})^\omega$), the resulting orbit equivalence relations are actually universal for any non-Archimedean TSI (resp. abelian) Polish group action.
\end{proof}

Now we move on to proving the theorems relating to topological obstructions to classification by non-Archimedean TSI actions. 

\begin{proposition}\label{prop:generic_ergodicity}
Suppose $E$ is an equivalence relation which is generically $E_\infty$-ergodic. Then $E$ is generically $\sim^T_\alpha$-ergodic, for any countable ordinal $\alpha$ and group tree $T$.
\end{proposition}

\begin{proof}

We first need to prove the following technical claim.
\begin{claim}\label{lem:squiggle_quotient}
For any group tree $T$, countable ordinal $\alpha$, $z \in \mathbb{R}^T$, and $n < \omega$,
\[ \sim^T_{\alpha, n} \upharpoonright [z]_{\sim^T_\alpha} \le_B E_\infty.\]
\end{claim}

\begin{proof}
Let $Y := \prod_{\beta < \alpha} \prod_{k \in \omega} 2^{T^{n+k}}$. Define a function $g : \mathbb{R}^T \rightarrow Y^{T^n}$ by
\[ g(\mathbf{x})(d)(\beta)(k)(e) = 1 \iff \exists f \in T_d, (\mathbf{x}, d \smallfrown f) \sim^T_\beta (\mathbf{z}, e).\]
There is a natural action of $T^n$ on $Y^{T^n}$ by shifts. We claim that $g$ is a Borel reduction from $\sim^T_{\alpha, n}$ restricted to $[\mathbf{z}]_{\sim^T_\alpha}$ to that orbit equivalence relation.

Suppose that $\mathbf{x} \sim^T_{\alpha, n} \mathbf{y}$. Then there is some $d \in T^n$ such that $(\mathbf{x}, 1^n) \sim_\alpha^T (\mathbf{y}, d)$. We'll see that $g(\mathbf{x})(1^n) = g(\mathbf{y})(d)$. Fix some arbitrary $\beta < \alpha$ and $k \in \omega$, and fix some $c \in T_d^k$ such that $(\mathbf{x}, 1^{n+k}) \sim^T_\beta (\mathbf{y}, d \smallfrown c)$. Then for any $e \in T^{n+k}$,
\begin{align*}
    &\exists f \in T_{1^n}^k, (\mathbf{x}, 1^n \smallfrown f) \sim^T_\beta (\mathbf{z}, e)\\
    \iff &\exists f \in T_{1^n}^k (\mathbf{x}, 1^{n+k}) \sim^T_\beta (\mathbf{z}, e \cdot (1^n \smallfrown f)^{-1})\\
    \iff &\exists f \in T_{1^n}^k (\mathbf{y}, d \smallfrown c) \sim^T_\beta (\mathbf{z}, e \cdot (1^n \smallfrown f)^{-1} \\
    \iff &\exists f \in T_{1^n}^k (\mathbf{y}, d \smallfrown (c \cdot f)) \sim^T_\beta (\mathbf{z}, e) \\
    \iff &\exists f \in T_{d}^k (\mathbf{y}, d \smallfrown f) \sim^T_\beta (\mathbf{z}, e)
\end{align*}
as desired.

Now suppose that $g(\mathbf{x})(1^n) = g(\mathbf{y})(d)$. We must show that for any $\beta < \alpha$ and $k \in \omega$, that $(\mathbf{x}, 1^n) \sim^T_{\beta, k} (\mathbf{y}, d)$. We know $\mathbf{x} \sim \mathbf{z}$, so for some $e \in T^{n+k}$ we have $(\mathbf{x}, 1^{n+k}) \sim^T_\beta (\mathbf{z}, e)$. Thus there exists some $f \in T_d^k$ such that $(\mathbf{y}, d \smallfrown f) \sim^T_\beta (\mathbf{z}, e)$ and so by transitivity we have $(\mathbf{x}, 1^{n+k}) \sim^T_\beta (\mathbf{y}, d \smallfrown f)$ as desired.
\end{proof}

Suppose $E$ is generically $E_\infty$-ergodic. Since $\sim^T_0$ is smooth, $E$ is generically $\sim^T_0$-ergodic. Furthermore, for a given ordinal $\alpha$, if $E$ is generically $\sim^T_{\alpha, n}$-ergodic for every $n \in \omega$, it follows by the countable closure of the meager ideal that $E$ is generically $\sim^T_{n+1}$-ergodic. Similarly, if $E$ is generically $\sim^T_\beta$-ergodic for every $\beta < \alpha$, with $\alpha$ countable, then it follows that $E$ is generically $\sim^T_\alpha$-ergodic.
\end{proof}

\topobstructionthm

\begin{proof}
Suppose $G$ is a non-Archimedean TSI Polish group and $X$ is a Polish $G$-space. Let $E$ be an equivalence relation on Polish space $Y$ such that $E$ is generically $E_\infty$-ergodic. Fix some Baire-measurable homomorphism $f : Y \rightarrow X$ from $E$ to $E^G_X$. We can assume that $G = G_T$ for some group tree $T$, and by Theorem \ref{th:universal_map}, we can assume that $X$ is some invariant $\mathbf{\Pi}^0_2$ subset of $\mathbb{R}^T$ with the relative topology. 
As in Claim 5.4 of \cite{Allison_Panagio_2020}, we can find a comeager subset $C \subseteq Y$ and a countable ordinal $\lambda$ such that for every $y \in C$, $[f(y)]_{\cong}$ is $\mathbf{\Pi}^0_\lambda$. 
In particular, by Proposition \ref{pr:isom_sim}, we have some countable ordinal $\lambda'$ such that $\cong$ and $\sim_{\lambda'}$ coincide on $f[C]$. 
By Proposition \ref{prop:generic_ergodicity}, we have a comeager subset $C' \subseteq C$ such that $f[C]$ is contained in a single $\sim_{\lambda'}$ class, and thus it is contained in a single $\cong$-class. 
\end{proof} 

\ccapplication

\begin{proof}
In \cite{Clemens_Coskey}, it is shown that if $E$ is generically-ergodic and $\Delta$ is countably-infinite, then $E^{[\Delta]}$ is generically $E_\infty$-ergodic. By Theorem \ref{th:top_obstruction}, it is generically-ergodic with respect to every orbit equivalence relation induced by a non-Archimedean TSI Polish group. In the case that $E$ is countable, every class in $E^{[\Delta]}$ is meager, and thus it is not classifiable by non-Archimedean TSI Polish groups.
\end{proof}

\begin{proposition}\label{prop:generic_ergodicity2}
Suppose $E$ is an equivalence relation which is generically $E_0$-ergodic. Then $E$ is generically $\sim^T_\alpha$-ergodic, for any countable ordinal $\alpha$ and abelian group tree $T$.
\end{proposition}

\begin{proof}
As in the proof of Claim \label{lem:squiggle_quotient}, $\sim^T_{\alpha, n} \upharpoonright [z]_{\sim^T_\alpha}$ is Borel-reducible to an action of $T^n$, an abelian group. By \cite{Gao_Jackson_2015}, it is thus Borel-reducible to $E_0$. Following the proof of Proposition \ref{prop:generic_ergodicity}, the generic ergodicity follows.
\end{proof}

Then we immediately establish following modification of Theorem \ref{th:top_obstruction}.

\topobstructionthmtwo

We finish by giving a more direct proof of the following result of Ding and Gao, and prove a ``higher" version of it.

\begin{corollary}
Suppose $E$ is an essentially-countable equivalence relation that is classifiable by a non-Archimedean abelian Polish group. Then $E$ is Borel-reducible to $E_0$ (i.e. is essentially hyperfinite).
\end{corollary}

\begin{proof}
In light of Corollary \ref{cor:upper_bound} and Proposition \ref{pr:fine_universal}, it is enough to see that $\sim^T_{0, <\omega}$ is essentially hyperfinite. We saw in the proof of Claim \ref{lem:squiggle_quotient} that each $\sim^T_{0, n}$ is induced by an action of $T^n$, a countable abelian group. Thus by \cite{Gao_Jackson_2015}, $\sim^T_{0, n}$ is essentially hyperfinite. The relation $\sim^T_{0, <\omega}$ is a direct sum of essentially hyperfinite equivalence relations, and thus it too is essentially hyperfinite.
\end{proof}

The original proof relied on an application of the Seventh Dichotomy Theorem, proved in \cite{Hjorth_Kechris_1997}. The dichotomy result used effective descriptive set theory in an essential way, while our arguments throughout this paper depend only on the fine analysis of Borel sets, and the Gao-Jackson result from \cite{Gao_Jackson_2015} on actions of countable abelian groups. Another straightforward proof of the Ding-Gao result, of a different nature, was independently given in \cite{Grebik_2020}, using the notion of \emph{$\sigma$-lacunary} actions. 

Finally, we state and prove the ``higher" analogue.

\classificationcor

\begin{proof}
As in the previous proof, it is enough to observe that $\sim^T_1$ is Borel-reducible to the product of the $\sim^T_{0, n}$, each of which is countable and thus Borel-reducible to $E_\infty$. In the case that $G_T$ is non-Archimedean abelian, each $\sim^T_{0, n}$ is induced by an action of $T^n$, which is a countable abelian group, and thus by the result \cite{Gao_Jackson_2015}, is hyperfinite. And thus $E$ is Borel-reducible to $E_0^\omega$.
\end{proof}



\newpage{}


