Borel equivalence relations induced by actions of tsi Polish groups

Recall that we denote as $E^{X}_{G}$ the orbit equivalence relation that is induced by a Borel action $G\curvearrowright X$ of a Polish group $G$ on a standard Borel space $X$, that is,

for every $x,y\in X$. The main technical tool that we use in this paper are variants of the ${\mathbbm{G}}_{0}$-dichotomy of Kechris, Solecki and Todorcevic [KST99]. Recall that the ${\mathbbm{G}}_{0}$-dichotomy characterizes Borel graphs of Borel chromatic number at most $\aleph_{0}$. That is, a Borel graph either admits a decomposition into at most countably many independent sets or a Borel homomorphism from the graph ${\mathbbm{G}}_{0}$, a canonical example of graph that does not admit such a decomposition. The elementary proof of this dichotomy is due to Miller [Mil12].

Our approach is profoundly influenced by [Kec92, Hjo00a, Hjo00b, Mila]. In particular, the results of Miller [Mila] are not only literally used as a part of the proof of our main result but the ideas are internally present throughout the paper.

The starting goal of this project, also suggested by Miller, was to understand Hjorth’s ${\mathbbm{E}}_{2}$-dichotomy [Hjo00a]. Recall that the equivalence relation ${\mathbbm{E}}_{2}$ is induced by the canonical action of the Banach space $\ell_{1}$ on ${\mathbbm{R}}^{{\mathbbm{N}}}$ and that this action is turbulent, i.e., ${\mathbbm{E}}_{2}$ does not admit classification by countable structures. Hjorth’s ${\mathbbm{E}}_{2}$-dichotomy states that a restriction of ${\mathbbm{E}}_{2}$ to any Borel subset of ${\mathbbm{R}}^{{\mathbbm{N}}}$ is either essentially countable or Borel bi-reducible with ${\mathbbm{E}}_{2}$. To prove this result one can use a variant of the ${\mathbbm{G}}_{0}$-dichotomy as follows. The dichotomy is applied to a family of oriented hypergraphs $\mathcal{H}_{V,W}$, where for every open neighborhoods $W\subseteq V$ of the identity in $G$ we define an oriented hypergraph $\mathcal{H}_{V,W}$ on $X$ by declaring $x\in X^{<\aleph_{0}}$ to be an edge if the consecutive elements are connected by elements in $W$ and the first and last element of $x$ are not connected by any element of $V$. Fixing $V$ and applying a variant of the ${\mathbbm{G}}_{0}$-dichotomy to a decreasing sequence of open neighborhoods of $1_{G}$ gives either a homomorphism from a canonical ${\mathbbm{G}}_{0}$-like object or a decomposition into $\mathcal{H}_{V,W}$-invariant sets. In the case of homomorphism one can use a refinement technique, see [Mila], to find a reduction from ${\mathbbm{E}}_{2}$. In the other case, it is not hard to see that the corresponding decomposition is made of sets that intersect each orbit in bounded and separated islands (or galaxies, aka grainy sets [Kan08, Chapter 15.2]). Formally, we make the following definition, see Section 4.

To understand why we derive essential countability in this case, it might be illustrative to make a detour and discuss Kechris’ result [Kec92] that says that every equivalence relation induced by a Borel action of a locally compact Polish group is essentially countable. As a first step in the original proof Kechris basically shows that every such action satisfies Property (IC). The reason why we can conclude in both examples that the equivalence relation is essentially countable is a combination of Property (IC) and the following notion. We say that an action $G\curvearrowright X$ admits a $G$-bounded topology if there is a compatible Polish topology $\tau$ on $X$ that makes the action continuous¹¹1Compatible topology that makes the action continuous is called a $G$-Polish topology. and such that for every vertex $x\in X$ there is an open neighborhood $\Delta$ of $1_{G}$ such that acting with group elements from $\Delta$ on $x$ does not approximate elements from different orbit, i.e., the $\tau$-closure of $\Delta\cdot x$ is a subset of the orbit of $x$. It is easy to see that if $G$ is compact and the action is continuous, then the topology is $G$-bounded. Also the canonical topology on ${\mathbbm{R}}^{{\mathbbm{N}}}$ is $\ell_{1}$-bounded for the canonical action of the Banach space $\ell_{1}$. The proof of Hjorth’s theorem and Kechris’ theorem is then finished by our first result

A natural question is to understand in which situation we can apply the aforementioned variant of the ${\mathbbm{G}}_{0}$-dichotomy for the oriented hypergraphs $\mathcal{H}_{V,W}$ and get similar results. It is not difficult to show that if we have a decomposition into $\mathcal{H}_{V,W}$-independent sets, then the action is not turbulent (this result hold for any Polish group). We obtain the following result.

The reason why Property (IC) implies classification by countable structures is intuitively clear, the corresponding Borel decomposition into $\mathcal{H}_{V,W}$-independent sets mimics the behavior of actions of non-archimedean groups. For this class Property (IC) holds trivially, and it is well-known that the corresponding orbit equivalence relations admit classification by countable structures. The other implication is more challenging. By a variant of the ${\mathbbm{G}}_{0}$-dichotomy we get a homomorphism from some canonical object. We use heavily the assumption that the group is tsi to do several refinements. Ultimately we obtain a Borel reduction from a Borel equivalence relation that is closely connected to tall lsc submeasures, or $c_{0}$-equalities, [Far01a, Far01b, Kan08]. These are known to be induced by turbulent actions and this can be used to show that $E^{X}_{G}$ cannot be classifiable by countable structures. Even though we are not able not show that tall lsc submeasures and $c_{0}$-equalities form a base for non-classification by countable structures, our result serves as an indication that this might be the case for equivalence relations that are induced by actions of tsi Polish groups. For general Borel equivalence relations, in particular, the ones induced by actions of general Polish groups, we do not have any intuition.

Next we turn our attention to our main result. We use results of Miller [Mila] to show that ${\mathbbm{E}}_{0}^{{\mathbbm{N}}}$ is a canonical obstacle for essential countability under the assumption that $E^{X}_{G}$ admits classification by countable structures. In the case of non-archimedean tsi Polish groups, the result was proved by Hjorth and Kechris [HK01]. Recently Miller found a proof that uses a variant of the ${\mathbbm{G}}_{0}$-dichotomy [Mila]. We manage to weaken the assumption to merely tsi Polish groups but we need to keep the assumption that the equivalence relation admits classification by countable structures. Note that without this assumption the situation is more complicated, e.g., ${\mathbbm{E}}_{2}$ is not essentially countable but there is no reduction from ${\mathbbm{E}}_{0}^{{\mathbbm{N}}}$ to ${\mathbbm{E}}_{2}$. In general, there is no reduction from ${\mathbbm{E}}_{0}^{{\mathbbm{N}}}$ if the corresponding action admits $G$-bounded topology.

The strategy for proving this result is to use the ${\mathbbm{G}}_{0}$-dichotomy two times. First, by Theorem 1.3 we get that Property (IC) holds, i.e., there is a Borel decomposition that mimics the behavior of non-archimedean groups. Second, we use the result of Miller [Mila] who found a variant of the ${\mathbbm{G}}_{0}$-dichotomy that characterizes $\sigma$-lacunarity (a formal strengthening of essential countability). If we get a Borel decomposition into independent sets in his result, then we conclude that the action is $\sigma$-lacunary, i.e., essentially countable. In the other case we get a homomorphism from some canonical object. Miller was able to refine the homomorphism to get a reduction from ${\mathbbm{E}}_{0}^{{\mathbbm{N}}}$ under the assumption that the group is non-archimedean. We show that his argument goes through under the weaker assumption of Property (IC).

A topological space $X$ is a Polish space if the underlying topology is separable and completely metrizable. A standard Borel space $X$ is a set endowed with a $\sigma$-algebra that is a $\sigma$-algebra of Borel sets for some Polish topology on $X$. We call such a Polish topology compatible. A topological group $G$ is a Polish group if the underlying topology is Polish. We denote the $\sigma$-ideal of meager sets on $G$ as $\mathcal{M}_{G}$. We use the category quantifiers $\exists^{*}$, $\forall^{*}$ in the standard meaning, e.g.,

where $U\subseteq G$ is an open set and $P$ is some property, see [Gao09].

A Borel action $G\curvearrowright X$ of a Polish group $G$ on a standard Borel space $X$ is an action that is additionally Borel measurable as a function from $G\times X$ to $X$. We write $(g,x)\mapsto g\cdot x$ for the evaluation of the action at particular elements $g\in G$ and $x\in X$. Similarly, we define $V\cdot x$ for any $V\subseteq G$ and $x\in X$. We denote as $E^{X}_{G}$ the induced equivalence relation and as $[x]_{E^{X}_{G}}$ the equivalence class, or orbit, of $x\in X$. A set $A\subseteq X$ is $G$-invariant if it is a union of equivalence classes of $E^{X}_{G}$. If $V\subseteq G$, then we set $(x,y)\in R^{X}_{V}$ if and only if $y\in V\cdot x$. It is a result of Becker and Kechris [BK96] that one can always find a compatible Polish topology on $X$ such that the action is continuous. Any such Polish topology is called a $G$-Polish topology. If such a topology is fixed we say that $G\curvearrowright X$ is a Polish $G$-space. For $x\in X$ and $A\subseteq X$ we set $G(x,A)=\{g\in G:g\cdot x\in A\}$.

A Borel equivalence relation $E$ on a standard Borel space $X$ is is an equivalence relation that is additionally a Borel subset of $X\times X$. We assume throughout the paper that the orbit equivalence relations of the form $E^{X}_{G}$ that we consider are always Borel equivalence relations. A Borel equivalence relation $E$ on $X$ is Borel reducible to a Borel equivalence $F$ on $Y$, in symbols $E\leq_{B}F$, if there is a Borel map $\varphi:X\to Y$ that is a reduction from $E$ to $F$.

We say that a Borel equivalence relation $E$ is essentially countable if it is Borel reducible to some countable Borel equivalence relation, see [Kec21].²²2A Borel equivalence relation is countable if the cardinality of each equivalence class is at most countable. In our setting, that is Borel equivalence relations inudeced by Polish group actions, this is equivalent to $\sigma$-lacunarity [Gre20].

We say that a Borel equivalence relation $E$ is classifiable by countable structures if it is Borel reducible to some equivalence relation induced by a Borel action of $S_{\infty}$, the permutation group of ${\mathbbm{N}}$. There are several other equivalent characterizations of classification by countable structures [Kan08, Theorem 12.3.3] or [Gao09].

The benchmark examples of Borel equivalence relations that we consider in this paper are ${\mathbbm{E}}_{0}$, ${\mathbbm{E}}_{2}$ and ${\mathbbm{E}}_{0}^{{\mathbbm{N}}}$. The equivalence relation ${\mathbbm{E}}_{0}$ on $2^{{\mathbbm{N}}}$ is defined as $(x,y)\in{\mathbbm{E}}_{0}$ if and only if $|\{n\in{\mathbbm{N}}:x(n)\not=y(n)\}|<\aleph_{0}$. The equivalence relation ${\mathbbm{E}}_{2}$ is defined in the Introduction. The equivalence relation ${\mathbbm{E}}_{0}^{{\mathbbm{N}}}$ is the countable product of ${\mathbbm{E}}_{0}$, that is, ${\mathbbm{E}}_{0}^{{\mathbbm{N}}}$ is the equivalence relation on $2^{{\mathbbm{N}}\times{\mathbbm{N}}}$ defined as $(x,y)\in{\mathbbm{E}}_{0}^{{\mathbbm{N}}}$ if and only if $|\{m\in{\mathbbm{N}}:x(n,m)\not=y(n,m)\}|<\aleph_{0}$ holds for every $n\in{\mathbbm{N}}$. We refer the reader to [Kan08] for more information about these particular examples.

Let $X$ be a Polish $G$-space, $V\subseteq G$, $U\subseteq X$ and $x\in X$. We introduce some notation that is connected to the definition of local orbit, see [Gao09, Section 10.2]. First, we define

the set of all $V$-jumps. Let $\mathcal{J}(x,V)=\left\{y\in\mathcal{J}(V):y_{0}=x\right\}$. Assuming now that $U$ and $V$ are open neighborhoods of $x\in X$ and the identity $1_{G}\in G$, respectively, we define the local orbit

That is, $\mathcal{O}(x,U,V)$ are those elements of $U$ that are reachable from $x$ by $V$-jumps within $U$.

Let $E$ be an equivalence relation on a Polish space $X$. We say that $E$ is generically $S_{\infty}$-ergodic if for every Polish $S_{\infty}$-space $Y$ and every Baire measurable homomorphism $\varphi:X\to Y$ from $E$ to $E^{Y}_{S_{\infty}}$ there is $y\in Y$ such that $\varphi^{-1}\left([y]_{E^{Y}_{S_{\infty}}}\right)$ is comeager in $X$.

A Polish group $G$ is tsi (states for two-sided invariant) if there is an open basis at $1_{G}$ made of conjugacy invariant open sets. That is, there is a sequence $(\Delta_{k})_{k\in{\mathbbm{N}}}$ of open neighborhoods of $1_{G}$ that is an open base and such that $g\cdot\Delta_{k}\cdot g^{-1}=\Delta_{k}$ for every $g\in G$ and $k\in{\mathbbm{N}}$. Equivalently, see [Gao09, Exercise 2.1.4], there is a compatible metric $d$ on $G$ that is two sided invariant, i.e., $d(g,h)=d(h^{-1}\cdot g,1_{G})=d(g\cdot h^{-1},1_{G})$ for every $g,h\in G$. It follows from [Gao09, Exercise 2.2.4] that such a metric $d$ is necessarily complete. We always assume that such a metric $d$ on $G$ is fixed and put $\Delta_{\epsilon}=\{g\in G:d(g,1_{G})<\epsilon\}$. We have $g\cdot\Delta_{\epsilon}\cdot g^{-1}=\Delta_{\epsilon}$ for every $\epsilon>0$ and $g\in G$. We abuse the notation and define $\Delta_{k}=\Delta_{\frac{1}{2^{k}}}$ for every $k\in{\mathbbm{N}}$. Note that $(\Delta_{k})_{k\in{\mathbbm{N}}}$ is a conjugacy invariant open base at $1_{G}$ such that $\Delta_{k+1}\cdot\Delta_{k+1}\subseteq\Delta_{k}$ and $\Delta_{k}=\Delta_{k}^{-1}$ for every $k\in{\mathbbm{N}}$.

We define the dihypergraphs that we use in this paper.

We formulate three versions of the ${\mathbbm{G}}_{0}$-dichotomy. First version is the original dichotomy and the other two are the versions that we use in this paper. We formulate the statements in bigger generality as it is done in [Milb].

For $s\in 2^{<{\mathbbm{N}}}$ define the graph

on $2^{\mathbbm{N}}$. Fix some dense collection $(s_{n})_{n\in{\mathbbm{N}}}\subseteq 2^{<{\mathbbm{N}}}$ such that $|s_{n}|=n$, i.e., $s_{n}\in 2^{n}$, for every $n\in{\mathbbm{N}}$. Here dense means that for every $s\in 2^{<{\mathbbm{N}}}$ there is $n\in{\mathbbm{N}}$ such that $s\sqsubseteq s_{n}$. Set ${\mathbbm{G}}_{0}=\bigcup_{n\in{\mathbbm{N}}}{\mathbbm{G}}_{s_{n}}$.

Next we formulate two versions of the ${\mathbbm{G}}_{0}$-dichotomy that we use in this paper. For the first one we need to recall some notation from [Mila]. Let $k_{n}\in{\mathbbm{N}}$ be such that $k_{0}=0$, $k_{n+1}\leq\max\{k_{m}:m\leq n\}+1$ for every $n\in{\mathbbm{N}}$ and for every $k\in{\mathbbm{N}}$ there are infinitely many $n\in{\mathbbm{N}}$ such that $k_{n}=k$. Fix $(s_{n})_{n\in{\mathbbm{N}}}\in 2^{{\mathbbm{N}}}$ such that $|s_{n}|=n$ for every $n\in{\mathbbm{N}}$ and $\{s_{n}:k_{n}=k\}$ is dense in $2^{<{\mathbbm{N}}}$ for every $k\in{\mathbbm{N}}$. Set

for every $k\in{\mathbbm{N}}$.

We define a generalization of ${\mathbbm{G}}_{0}$ and ${\mathbbm{E}}_{0}$ for special class of finitely branching trees. We say that a tree $T\subseteq{\mathbbm{N}}^{<{\mathbbm{N}}}$ is finitely uniformly branching if there is a sequence $(\ell^{T}_{m})_{m\in{\mathbbm{N}}}$ of natural numbers such that $\ell^{T}_{m}\geq 2$ for every $m\in{\mathbbm{N}}$ and we have

for every $s\in T$. If $T$ is a tree and $s\in T$, then we define $T_{s}=\{t\in{\mathbbm{N}}^{<{\mathbbm{N}}}:s^{\frown}t\in T\}$. Note that $T_{s}=T_{t}$ whenever $t,s\in T$ and $|t|=|s|$. We denote as $[T]\subseteq{\mathbbm{N}}^{{\mathbbm{N}}}$ the set of all branches through $T$, i.e., $\alpha\in[T]$ if and only if $\alpha\upharpoonright m\in T$ for every $m\in{\mathbbm{N}}$.

The equivalence relation ${\mathbbm{E}}^{T}_{0}$ on $[T]$ is defined as

where $\alpha,\beta\in[T]$. Note that in the case when $T=2^{<{\mathbbm{N}}}$ we have ${\mathbbm{E}}^{T}_{0}={\mathbbm{E}}_{0}$.

We describe a canonical way how to find Borel pseudometrics. Recall that if $G$ is a tsi Polish group, then $d$ is a fixed compatible two-sided invariant metric on $G$.

Every Borel pseudoemtric ${\bf d}$ on $[T]$ defines a Borel equivalence relation $F_{\bf d}$ on $[T]$ as

Note that in the case of Proposition 2.10 we have that $F_{{\bf d}}=\left(\varphi^{-1}\times\varphi^{-1}\right)(E^{X}_{G})$.

We describe the family of Borel equivalence relations that will serve as a base under Borel reducibility for non-classification by countable structures in the proof of Theorem 1.3. To that end we recall definitions of two types of Borel equivalence relations that are well-studied [Far01a, Far01b] and [Kan08, Chapters 3 and 15]. We denote the power set of ${\mathbbm{N}}$ as $\mathcal{P}({\mathbbm{N}})$.

A map $\Theta:\mathcal{P}({\mathbbm{N}})\to[0,+\infty]$ is a lsc submeasure if $\Theta(\emptyset)=0$, $\Theta(M\cup N)\leq\Theta(M)+\Theta(N)$ whenever $M,N\in\mathcal{P}({\mathbbm{N}})$, $\Theta(\{m\})<+\infty$ for every $m\in{\mathbbm{N}}$ and

for every $M\in\mathcal{P}({\mathbbm{N}})$. We say that $\Theta$ is tall if $\lim_{m\to\infty}\Theta(\{m\})=0$.

Let $\Theta$ be a tall lsc submeausre. Then the equivalence relation $E_{\Theta}$ on $2^{{\mathbbm{N}}}$ is defined as

for every $x,y\in 2^{{\mathbbm{N}}}$. Similarly as in Theorem 2.11, we have that $E_{\Theta}$ is non-meager if and only if $E_{\Theta}=2^{{\mathbbm{N}}}\times 2^{{\mathbbm{N}}}$. We refer the reader to [Kan08, Chapter 3] for more information about lsc submeasures and their connection to Borel reducibility.

A sequence of finite metric spaces $((Z_{m},\mathfrak{d}_{m}))_{m\in{\mathbbm{N}}}$ is called non-trivial if

where $r(Z,\mathfrak{d})=\max\mathfrak{d}$ and $j(Z,\mathfrak{d})$ is the minimal $\epsilon>0$ such that there is $\ell\in{\mathbbm{N}}$ and a sequence $(z_{0},\dots z_{l})$ that contains every element of $Z$ and satisfies $\mathfrak{d}(z_{i},z_{i+1})<\epsilon$ for every $i<l$.