We show that every essentially countable orbit equivalence relation induced by a continuous action of a Polish group on a Polish space is $σ$-lacunary. In combination with [Invent. Math.201 (1), 309-383, 2015] we obtain a straightforward proof of the result from [Adv. Math.307, 312-343,2017] that every essentially countable equivalence relation that is induced by an action of abelian non-archimedean Polish group is essentially hyperfinite.
Under a mild definability assumption, we characterize the family of Borel actions $Γ\curvearrowright X$ of tsi Polish groups on Polish spaces that can be decomposed into countably-many actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of $1_Γ$. In the special case that $Γ$ is non-archimedean, it follows that there is such a decomposition if and only if there is no continuous embedding of $\mathbb{E}_0^{\mathbb{N}}$ into $E_Γ^X$.
Let $X$ be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group $G$ on $X.$ If $G$ and $X$ are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups.
We extend the result of Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the general case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group action.
We define some coding of Borel sets in admissible sets. Using this we generalize certain results from model theory involving admissible sets to the case of continuous actions of closed permutation groups on Polish spaces. In particular we obtain counterparts of Nadel's theorems about relationships between Scott sentences and admissible sets.
Let $Γ$ and $Δ$ be sufficiently distinct countable groups. We show that there is an orbit equivalence relation $E$, induced by an action of the Polish wreath product group $Γ\wrΓ$, so that $E$ is generically $F$-ergodic for any orbit equivalence relation $F$ induced by an action of $Δ\wrΔ$. More generally, we establish generic ergodicity between $Γ$-jumps and the iterated $Δ$-jumps, answering a question of Clemens and Coskey. The proofs follow a translation between Borel homomorphisms and definable pins.
In this paper we consider non-archimedean abelian Polish groups whose orbit equivalence relations are all Borel. Such groups are called tame. We show that a non-archimedean abelian Polish group is tame if and only if it does not involve $\Z^ω$ or $(\Z(p)^{<ω})^ω$ for any prime $p$. In addition to determining the structure of tame groups, we also consider the actions of such groups and study the complexity of their orbit equivalence relations in the Borel reducibility hierarchy. It is shown that if such an orbit equivalence relation is essentially countable, then it must be essentially hyperfin…
Let G be a closed subgroup of the group of all permutations of a countably infinite set. Let X be a Polish G-space with a countable basis A of clopen sets. Each x from X defines a characteristic function f on A by f(U)=1 iff x belongs to U (where U is from A). We consider computable complexity of f and some related questions.
We study when a continuous isometric action of a Polish group on a complete metric space is, or can be, transitive. Our main results consist of showing that certain Polish groups, namely $\mathrm{Aut}^*(μ)$ and $\mathrm{Homeo}^+[0,1]$, such an action can never be transitive (unless the space acted upon is a singleton). We also point out "circumstantial evidence" that this pathology could be related to that of Polish groups which are not closed permutation groups and yet have discrete uniform distance, and give a general characterisation of continuous isometric action of a Roeckle-precompact Po…
Hjorth has introduced a Scott analysis for general Polish group actions, and has asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank - namely, the orbit equivalence relation is Borel if and only if Hjorth ranks are bounded. We present the principles of Hjorth analysis and Hjorth rank, and answer Hjorth's question positively. From that we get a positive answer to a conjecture due to Hjorth - for every limit ordinal $α$ , the set of elements whose orbit is of complexity less than $α$ is a Borel set. We then show Nadel's theorem for Hjorth rank - th…
We study Borel equivalence relations induced by Borel actions of tsi Polish groups on standard Borel spaces. We characterize when such an equivalence relation admits classification by countable structures using a variant of the $\mathbb G_0$-dichotomy. In particular, we find a class that serves as a base for non-classification by countable structures for these equivalence relations under Borel reducibility. We use this characterization together with the result of [B. D. Miller, to appear in the Journal of Mathematical Logic] to show that if such an equivalence relation admits classification by…
We analyse logic actions of Polish groups which arise in continuous logic. We extend the generalised model theory of H.Becker to the case of Polish G-spaces when G is an arbitrary Polish group.
We study continuous actions of Polish groups on Polish spaces. We develop Scott analysis introduced by Hjorth for studying orbit equivalence relations. We define eventually open actions and prove that this property characterizes the actions endowed with a complete system of hereditarily countable invariant structures.
We study non-nesting actions of groups on real trees. We prove some fixed point theorems for such actions under the assumption that groups are Polish and have comeagre conjugacy classes.
Let $X = \left\{P \in [0,1]^{\bf N} : \left(\forall ν\in {\bf N} \right) \left(P \left(\{ν\} \right) > 0 \right) \wedge \sum\limits_{ν= 0}^{\infty} P \left(\{ν\} \right) = 1 \right\} $ be the Polish space of probability measures on ${\bf N}$, each of which assigns positive probability to every elementary event, while for any $P \in X$, let $Γ_{P} = \left\{ξ\in L^{1}({\bf N}, P) : \left(\forall ν\in {\bf N} \right) \left(ξ(ν) > 0 \right) \wedge \sum\limits_{ν= 0}^{\infty} ξ(ν) P \left(\{ν\} \right) = 1 \right\} $ and let $Φ_{P} : Γ_{P} \ni ξ\mapsto Φ_{P}(ξ) \in X$ be defined by the relation $\l…
We show that a {\it Borel} action of a Polish group on a standard Borel space is Borel isomorphic to a {\it continuous} action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture. We establish the existence of universal actions for any given Polish group, extending a result of Mackey and Varadarajan for the locally compact case. We prove an analog of Tarski's theorem on paradoxical decompositions by showing that the existence of an inva…
We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $τ$ on $X_G$ such that $(X_G, τ)$ is Polish and the quotient Borel structure on $X_G$ is equal to $Borel(X_G,τ)$. To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups.
We show that every abelian Polish group is the topological factor-group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the unitary group.
We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed under the Fubini product modulo the ideal Fin of all finite sets, and some other operations. Our second main result shows that T_2, the equivalence relation called ``the equality of countable sets of the reals'', is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with G_δclasses, …
We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims at measuring the complexity of classifying the objects of a category in a constructive and functorial way. In the particular case of principal groupoids such a notion coincide with the usual Borel complexity of equivalence relations. Our main result is that on one hand for Polish groupoids with essentially treeable orbit equivalence relation, functorial Borel complexity coincides with the Borel complexity of the associated orbit equivalence relation. On the other hand for every countable e…
Becker and Kechris showed that if a Polish group G acts continuously on a Polish space X, then for any invariant Borel set B we can change the topology on X so that B becomes open, the Borel structure is preserved, and the action continues to be continuous. In this brief paper a short proof is presented for their theorem. The method also gives optimal bounds in terms of minimizing the change to the original topology.
We study topological groups that can be defined as Polish, pro-countable abelian groups, as non-archimedean abelian groups or as quasi-countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups such that all orbit equivalence relations induced by their continuous actions on Polish spaces are Borel, and relatively tame groups $G$, i.e., groups such that every diagonal action $α\times β$ of $G$ induces a Borel orbit equivalence relation, provided that the actions $α$, $β$ of $…
Let $G$ be a locally compact Polish group. A metrizable $G$-flow $Y$ is called model-universal if by considering the various invariant probability measures on $Y$, we can recover every free action of $G$ on a standard Lebesgue space up to isomorphism. Weiss has shown that for countable $G$, there exists a minimal, model-universal flow. In this paper, we extend this result to all locally compact Polish groups.
This paper considers "definable cardinalities" arising from Polish group actions. The first part of the paper answers a question of Becker-Kechris by showing that under suitable determinacy assumptions in ZF+DC, every action by a Polish group on a metric space is either as complicated as the Vitali equivalence relation or no more complicated than the equality relation on bounded subsets of \aleph_1. The same argument shows that if hereditarily wellorderabel sets can be used as complete invariants, then elements in HC will do. The second half gives a new proof of Howard Becker's recent theorem …
We study several distinguished function algebras on a Polish group $G$, under the assumption that $G$ is Roelcke precompact. We do this by means of the model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate the dynamics of $\aleph_0$-categorical metric structures under the action of their automorphism group. We show that, in this context, every strongly uniformly continuous function (in particular, every Asplund function) is weakly almost periodic. We also point out the correspondence between tame functions and NIP formulas, deducing that the isometry group of the Urys…
For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P := \{ gC : g \in G, C \in P \}$ generates the Borel $σ$-algebra of $X$. For $G = Z$, the Kolmogorov--Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite entropy. It was asked by Benjamin Weiss in the late 80s whether the nonexistence of any invariant probability measure guarantees the existence of a finite generator. We…
We prove that every open $σ$-locally Polish groupoid $G$ is Borel equivalent to the groupoid of models on the Urysohn sphere $\mathbb{U}$ of an $\mathcal{L}_{ω_1ω}$-sentence in continuous logic. In particular, the orbit equivalence relations of such groupoids are up to Borel bireducibility precisely those of Polish group actions, answering a question of Lupini. Analogously, every non-Archimedean (i.e., every unit morphism has a neighborhood basis of open subgroupoids) open quasi-Polish groupoid is Borel equivalent to the groupoid of models on $\mathbb{N}$ of an $\mathcal{L}_{ω_1ω}$-sentence in…
Given a continuous and isometric action of a Polish group $G$ on an adequate Polish topometric space $(X,τ,ρ)$ and $x \in X$, we find a necessary and sufficient condition for $\overline{Gx}^ρ$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.
We introduce the notion of $(Γ,E)$-determinacy for $Γ$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2=\{0,1\}$ or $S=ω$. We show that for all shift actions by countable groups $G$, and any "reasonable" pointclass $Γ$, that $(Γ,E_G)$-determinacy implies $Γ$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^\mathbb{Z}$.
A Polish group $G$ is tame if for any continuous action of $G$, the corresponding orbit equivalence relation is Borel. When $G = \prod_n Γ_n$ for countable abelian $Γ_n$, Solecki (1995) gave a characterization for when $G$ is tame. Ding and Gao (2017) showed that for such $G$, the orbit equivalence relation must in fact be potentially $\mathbfΠ^0_6$, while conjecturing that the optimal bound could be $\mathbfΠ^0_3$. We show that the optimal bound is $D(\mathbfΠ^0_5)$ by constructing an action of such a group $G$ which is not potentially $\mathbfΠ^0_5$, and show how to modify the analysis of Di…
We show that all groups of a distinguished class of «large» topological groups, that of Roelcke precompact Polish groups, have Kazhdan's Property (T). This answers a question of Tsankov and generalizes previous results by Bekka (for the infinite-dimensional unitary group) and by Evans and Tsankov (for oligomorphic groups). Further examples include the group $\operatorname{Aut}(μ)$ of measure-preserving transformations of the unit interval and the group $\operatorname{Aut}^*(μ)$ of non-singular transformations of the unit interval. More precisely, we prove that the smallest cocompact normal sub…
We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in $\mathbb{R}^3$ using Baire measurable pieces. We also obtain a Baire category solution to the dynamical…
We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product topology. We show, among other results, that for every non-locally compact, abelian quasi-countable group G there exists a closed subgroup L of G, and a closed, non-locally compact subgroup K of G/L which is a direct product of discrete, countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L are closed sub…
Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of L_{ω_1 ω}) is far from complete (see [5, 2]). In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivale…
A $*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $*$-reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish $G$-spaces which is generically preserved under $*$-reductions. For every natural number $n$ we define a free action of $S_{\infty}$ whose dimension is $n$ on every invariant Baire measurable non-meager set. We also show that the $S_{\infty}$-space which induc…
We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\Z$-actions, are coanalytic non-Borel inside $\Homeo(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside $\Homeo(X)$.
We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbfΣ^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth--Kechris about discrete structures. As a different application, we also give a new proof of Kechr…
Let $Γ$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^Γ)$ to denote the free part of the Bernoulli shift action of $Γ$ on $k^Γ$. Seward and Tucker-Drob showed that there exists a free subshift $\mathcal{S} \subseteq \mathrm{Free}(2^Γ)$ such that every free Borel action of $Γ$ on a Polish space admits a Borel $Γ$-equivariant map to $\mathcal{S}$. Here we generalize this result as follows. Let $\mathcal{S}$ be a subshift of finite type (for example, $\mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $Γ$ with some finite number of colo…
We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.
We show that if $G$ is a non-archimedean, Roelcke precompact, Polish group, then $G$ has Kazhdan's property (T). Moreover, if $G$ has a smallest open subgroup of finite index, then $G$ has a finite Kazhdan set. Examples of such $G$ include automorphism groups of countable $ω$-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of $G…
The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $Γ$ in Polish groups $G$, i.e. those elements in the Polish space $\mathrm{Rep}(Γ,G)$ of all representations of $Γ$ in $G$, whose orbit under the conjugation action of $G$ on $\mathrm{Rep}(Γ,G)$ is comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or $K_n$-free graphs, and we show its connections with Ribes-Zalesski-like properties of the acting groups. We prove that $\mathbb{N}$ has a generic representation in …
Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $μ$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $Γ$, there is a Borel sequence $(\cdot_r)_{r \in \mathbb{R}}$ of free actions of $Γ$ on $X$, generating subequivalence relations $E_r$ of $E$ with respect to which $μ$ is ergodic, with the further property that $(E_r)_{r \in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $μ$-reducibility. In particular, we show that if…
It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of $(\mathbb Q,+)$ by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring $\mathfrak A$ of finite adèles. Similarly, S. Solecki has proved that there is a comeagre set of mutually conjugate iso…
In the spirit of Hjorth's turbulence theory, we introduce "unbalancedness": a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two side invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace $C^*$-algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we s…
