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Countable Sections for Locally Compact Group Actions. II

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Alexander Kechris at California Institute of Technology
Alexander Kechris
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Abstract

It has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

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... The former was studied by Hjorth [Hjo00b], the theory of turbulence provides a dynamical obstacle for a continuous action of a Polish group to be classifiable by countable structures. The latter stems from the work of Kechris [Kec92], is ultimately connected with the theory of countable Borel equivalnce relations [Kec21], and some recent progress shows that this notion is equivalent to a geometrical notion of σ-lacunarity [Gre20] introduced and characterized by Miller in [Mila]. ...
... 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. ...
... 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). ...
Borel equivalence relations induced by actions of tsi Polish groups
Preprint
  • Jul 2021
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 G0\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 countable structures but it is not essentially countable, then the equivalence relation E0N=E3{\mathbb E}^{\mathbb N}_0=\mathbb E_3 Borel reduces to it.
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... The following definitions and theorems are useful. For proofs of these, we refer the reader to [12] or §5.4 of [8]. ...
... By a Borel selector for E, we mean a Borel function s : X → X such that for each x, y ∈ X, s(x)Ex and xEy ⇒ s(x) = s(y) Theorem 2.14. (Kechris, [12]) Let E be an equivalence relation on a standard Borel space X. Then E has a Borel selector iff E is smooth and idealistic. ...
... Theorem 2.16. (Kechris, [12]) Suppose G is a locally compact polish group acting continuously on a Polish space X. Then there is a Borel set Y ⊆ X which contains at least one and at most countably many points of each G-orbit (i.e., each E X Gequivalence class). ...
Abelian group actions and hypersmooth equivalence relations
Preprint
  • Oct 2019
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.
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... One can think of a cross section as being a discrete version of the ambient equivalence relation. A theorem of A. S. Kechris [Kec92] shows that all Borel flows admit cross sections. The notion of a cross section can be further strengthened by requiring cocompactness. ...
... For every > 0 we pick a Borel map α : R d → Q d such that r −α ( r) < for all r ∈ R d . By a theorem of Kechris [Kec92], there exists a B(2)-lacunary cross section C ⊆ X. Restriction E C of the orbit equivalence relation onto C is hyperfinite (see [JKL02, Theorem 1.16]), and one may therefore represent E C as an increasing union of finite equivalence relations: ...
On time change equivalence of Borel flows
Preprint
  • Sep 2016
This paper addresses the notion of time change equivalence for Borel multidimensional flows. We show that all free flows are time change equivalent up to a compressible set. An appropriate version of this result for non-free flows is also given.
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... Theorem 1.1 is deduced immediately from a theorem in descriptive set theory, that is, the existence of lacunary sections [8]. This will be discussed in Section 3. In order to be self-contained and to present the result to probabilists, a direct proof of Theorem 1.1 will also be given in Section 4. This proof is a simplification of the existence of lacunary sections in the special case of random measures on R d . ...
... It is called a lacunary section if there is a neighborhood U of the identity of G such that for every s ∈ S, one has (U · s) ∩ S = {s}. The following is a special case of Theorem 3.10 of [9], which is given originally in [8]: We will deduce Theorem 1.1 from the above theorem. ...
On the Existence of Balancing Allocations and Factor Point Processes
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Full-text available
  • Mar 2023
In this article, we show that every stationary random measure on R^d that is essentially free (i.e., has no symmetries a.s.) admits a point process as a factor. As a result, we improve the results of Last and Thorisson (2022) on the existence of a factor balancing allocation between ergodic pairs of stationary random measures Φ and Ψ with equal intensities. In particular, we prove that such an allocation exists if Φ is diffuse and either (Φ, Ψ) is essentially free or Φ assigns zero measure to every (d − 1)-dimensional affine hyperplane. The main result is deduced from an existing result in descriptive set theory, that is, the existence of lacunary sections. We also weaken the assumption of being essentially free to the case where a discrete group of symmetries is allowed.
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... The first gives a sufficient condition for a non-locally compact Polish group to have a Borel action on a standard Borel space with associated equivalence relation not essentially countable. It is a special case of [FR85, Theorem A] (see also [Ke92, Recall that an action of a group G on a set X is free if for every g = 1 and every x, g · x = x. Also for any action of G on X we denote by F (X) its free part defined by ...
... Proof of Application 3. The implication from right to left is immediate by Theorem 4.1 and [Ke92]. For the left-to-right implication, suppose that G is not locally compact and fix an open neighborhood of the identity, V . ...
On Polish groups admitting non-essentially countable actions
Article
  • Jan 2022
  • ERGOD THEOR DYN SYST
It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.
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... Similarly to the compact case, it is a theorem of Kechris [Ke92], that every orbit equivalence relation induced by any locally compact Polish group G is essentially countable. The question of whether this theorem too admits a converse was raised in [Ke92]; see also [Ke19, Problem 3.15]: ...
... Similarly to the compact case, it is a theorem of Kechris [Ke92], that every orbit equivalence relation induced by any locally compact Polish group G is essentially countable. The question of whether this theorem too admits a converse was raised in [Ke92]; see also [Ke19, Problem 3.15]: ...
On Polish groups admitting non-essentially countable actions
Preprint
  • Sep 2019
It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-Archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.
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... The orbit equivalence relation induced by a Polish group action is idealistic (see [20] and also [16,Theorem 5.4.10]). The analogous result holds for partial actions. ...
... A theorem of Kechris says that if E is an equivalence relation on a Polish space, then E has a Borel selector iff it is smooth and idealistic ( [20,Theorem 2.4], see also [16,Theorem 5.4.11]). Using this fact and Proposition 3.4 we immediately obtain the following generalization of a theorem of Burgess [16,Corollary 5.4.12]. ...
Borel Globalizations of Partial Actions of Polish Groups
Article
Full-text available
  • Aug 2018
  • ARCH MATH LOGIC
We show that the enveloping space XGX_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\tau on XGX_G such that (XG,τ)(X_G, \tau) is Polish and the quotient Borel structure on XGX_G is equal to Borel(XG,τ)Borel(X_G,\tau). 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.
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... One can think of a cross section as being a discrete version of the ambient equivalence relation. A theorem of A. S. Kechris [Kec92] shows that all Borel flows admit cross sections. The notion of a cross section can be further strengthened by requiring cocompactness. ...
... For every > 0 we pick a Borel map α : R d → Q d such that r −α ( r) < for all r ∈ R d . By a theorem of Kechris [Kec92], there exists a B(2)-lacunary cross section C ⊆ X. Restriction E C of the orbit equivalence relation onto C is hyperfinite (see [JKL02, Theorem 1.16]), and one may therefore represent E C as an increasing union of finite equivalence relations: ...
On time change equivalence of Borel flows
Article
  • Sep 2016
This paper addresses the notion of time change equivalence for Borel multidimensional flows. We show that all free flows are time change equivalent up to a compressible set. An appropriate version of this result for non-free flows is also given.
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... For example, S.Solecki [16] proved that a Polish group G is non-compact if and only if there exists a continuous action α of G such that E 0 is reducible to E α (where E 0 is the relation of eventual identity on the space 2 N of all 0 − 1 sequences.) Even more interestingly, A.Kechris [10] proved that if G is locally compact, then every orbit equivalence relation induced by G is reducible to an equivalence relation with countable classes. A natural question arises whether the converse holds. ...
... Proof. The implication from left to right follows from [10]. To show the other implication, fix K n , n ∈ N, and L as in Theorem 17. ...
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... Following [12] we can find B ⊂ X Borel meeting each orbit in a countable set such that for any compact neighborhood V ⊂ H of the identity and x ∈ B, ...
... This action is free and Borel, hence Ec is treeable. Applying [12] once more we obtain a complete section C0 ⊂ B1 for the action of ...
Treeable equivalence relations
Article
  • Jul 2012
There are continuum many ≤ B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group and hence, there are 2 α0 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility.
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... • ( [Kec92]) If G is a Polish locally compact group and X is a Borel G-space, then the orbit equivalence relation of G X admits classification by countable structures. ...
Non-classification of Cartan subalgebras for a class of von Neumann algebras
Preprint
  • Oct 2017
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II1_1 factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II1_1 factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II1_1 factor up to unitary conjugacy are not classifiable by countable structures, and deduce that the same holds for any McDuff II1_1 factor with at least one Cartan subalgebra.
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... Proposition 4.1 (Kechris [Kec92]). Suppose G is a locally-compact Polish group and a : G X is a Borel action on a standard Borel space. ...
Countable Borel treeable equivalence relations are classifiable by $\ell_1
Preprint
  • May 2023
Gao and Jackson showed that any countable Borel equivalence relation (CBER) induced by a countable abelian Polish group is hyperfinite. This prompted Hjorth to ask if this is in fact true for all CBERs classifiable by (uncountable) abelian Polish groups. We describe reductions involving free Banach spaces to show that every treeable CBER is classifiable by an abelian Polish group. As there exist treeable CBERs that are not hyperfinite, this answers Hjorth's question in the negative. On the other hand, we show that any CBER classifiable by a countable product of locally compact abelian Polish groups (such as Rω\mathbb{R}^\omega) is indeed hyperfinite. We use a small fragment of the Hjorth analysis of Polish group actions, which is Hjorth's generalization of the Scott analysis of countable structures to Polish group actions.
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... All non-smooth orbit equivalence relations produced by free Borel R-flows are Borel isomorphic to each other [14,Theorem 3]. Theorem 2.6 will show that this orbit equivalence relation can also be generated by a free action of any non-compact and nondiscrete Polish group. ...
Katok's special representation theorem for multidimensional Borel flows
Preprint
  • Feb 2023
Katok's special representation theorem states that any free ergodic measure-preserving Rd\mathbb{R}^{d}-flow can be realized as a special flow over a Zd\mathbb{Z}^{d}-action. It provides a multidimensional generalization of the "flow under a function" construction. We prove the analog of Katok's theorem in the framework of Borel dynamics and show that, likewise, all free Borel Rd\mathbb{R}^{d}-flows emerge from Zd\mathbb{Z}^{d}-actions through the special flow construction using bi-Lipschitz cocycles.
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... The situation is interesting already when Γ is a "well-understood" countable abelian group, such as the group of integers Z; the direct sum n∈ω Z p = Z <ω p of countably many copies of Z p , where Z p is the cyclic group of size p; the direct sum p prime Z p ; or the quasi-cyclic p-group Z(p ∞ ), for a prime number p (this group is isomorphic to z ∈ C : ∃n(z p n = 1) ). (1) (Kechris [Kec92]) If G is a locally compact group, then any orbit equivalence relation induced by a continuous action of G is Borel reducible to an orbit equivalence relation induced by a countable discrete group. 1 (2) (Gao-Jackson [GJ15]) Suppose Γ is a countable discrete group. If Γ is abelian then any orbit equivalence relation induced by Γ is Borel reducible to E 0 . 2 It follows that for any infinite countable discrete groups Γ, ∆, any orbit equivalence relation induced by Γ is Borel reducible to one induced by ∆. ...
Strong ergodicity for the Gamma-jump operator and for actions of Polish wreath products
Preprint
  • Mar 2022
Let Γ\Gamma and Δ\Delta 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≀Γ\Gamma\wr\Gamma, so that E is generically F-ergodic for any orbit equivalence relation F induced by an action of Δ≀Δ\Delta\wr\Delta. More generally, we establish generic ergodicity between Γ\Gamma-jumps and the iterated Δ\Delta-jumps, answering a question of Clemens and Coskey. The proofs follow a translation between Borel homomorphisms and definable pins.
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... We present here the notion of Borel-definable set and Borel-definable group as in [Lup20b,Lup20a]. We begin with recalling the notion of idealistic equivalence relation from [Kec94]; see also [Gao09,Definition 5.4.9] and [KM16]. We will consider as in [Lup20b] a slightly more generous notion. ...
(Looking For) The Heart of Abelian Polish Groups
Preprint
  • Feb 2022
We prove that the category M\mathcal{M} of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A\mathcal{A} of abelian Polish groups in the sense of Beilinson--Bernstein--Deligne and Schneiders. Thus, M\mathcal{M} is an abelian category containing A\mathcal{A} as a full subcategory such that the inclusion functor A→M\mathcal{A}\rightarrow \mathcal{M} is exact and finitely continuous. Furthermore, M\mathcal{M} is uniquely characterized up to equivalence by the following universal property: for every abelian category B\mathcal{B}, a functor A→B\mathcal{A}\rightarrow \mathcal{B} is exact and finitely continuous if and only if it extends to an exact (and necessarily finitely continuous) functor M→B\mathcal{M}\rightarrow \mathcal{B}. We provide similar descriptions of the left heart of the following categories: non-Archimedean abelian Polish groups; (non-Archimedean) Polish G-modules, for a given Polish group or Polish ring G; Fr\'{e}chet spaces over K for a given separable complete non-Archimedean valued field K.
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... The equivalence relation E 1 is defined on R N by x E 1 y ⇐⇒ ∃n∀m > n(x(m) = y(m)). Kechris and Louveau [KL97], extending [Kec92], proved that E 1 is not Borel reducible to any orbit equivalence relation induced by a Polish group action. (In particular, E 1 is not classifiable by countable structures.) ...
Classifying invariants for E1E_1: A tail of a generic real
Preprint
  • Dec 2021
Let E be an analytic equivalence relation on a Polish space. We introduce a framework for studying the possible "reasonable" complete classifications and the complexity of possible classifying invariants for E, such that: (1) the standard results and intuitions regarding classifications by countable structures are preserved in this framework; (2) this framework respects Borel reducibility; (3) this framework allows for a precise study of the possible invariants of certain equivalence relations which are not classifiable by countable structures, such as E1E_1. In this framework we show that E1E_1 can be classified, with classifying invariants which are κ\kappa-sequences of E0E_0-classes where κ=b\kappa=\mathfrak{b}, and it cannot be classified in such a manner if κ<add(B)\kappa<\mathbf{add}(\mathcal{B}). These results depend on analyzing the following sub-model of a Cohen real extension, introduced by Kanovei-Sabok-Zapletal (2013) and Larson-Zapletal (2020). Let <cn: n<ω>\left<c_n:\,n<\omega\right> be a generic sequence of Cohen reals, and define the tail intersection model M=⋂n<ωV[<cm: m≥n>].M=\bigcap_{n<\omega}V[\left<c_m:\,m\geq n\right>]. An analysis of reals in M will provide lower bounds for the possible invariants for E1E_1. We also extend the characterization of turbulence from Larson-Zapletal (2020) in terms of intersection models.
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... An important feature of locally compact group actions is the fact that they all admit measurable cross sections. This was proved by J. Feldman, P. Hahn, and C. Moore in [FHM78], whereas a Borel version of the result was obtained by A. S. Kechris in [Kec92]. ...
\mathrm{L}^1$ full groups of flows
Preprint
  • Aug 2021
We introduce the concept of an L1\mathrm{L}^{1} full group associated with a measure-preserving action of a Polish normed group on a standard probability space. Such groups are shown to carry a natural separable complete metric, and are thus Polish. Our construction generalizes L1\mathrm{L}^{1} full groups of actions of discrete groups, which have been studied recently by the first author. We show that under minor assumptions on the actions, topological derived subgroups of L1\mathrm{L}^{1} full groups are topologically simple and - when the acting group is locally compact and amenable - are whirly amenable and generically two-generated. For measure-preserving actions of the real line (also known as measure-preserving flows), the topological derived subgroup of an L1\mathrm{L}^{1} full groups is shown to coincide with the kernel of the index map, which implies that L1\mathrm{L}^{1} full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. The latter is in a striking contrast to the case of Z\mathbb{Z}-actions, where the number of topological generators is controlled by the entropy of the action.
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... Kechris in [8] proves the following. ...
Minimal model-universal flows for locally compact Polish groups
Article
  • Aug 2021
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.
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... Examples include the case where N is the n-dimensional Q-vector space or the complete n-branching tree. It is easy to see that this additional assumption forces P = Aut(N ) to be locally compact and, in turn, E(P ) to be essentially countable [Kec92]. From this point of view, bounded algebraic dimension can be seen as a partial weakening of local compactness. ...
Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension
Preprint
Full-text available
  • May 2021
The algebraic dimension of a Polish permutation group Q≤Sym(N)Q\leq \mathrm{Sym}(\mathbb{N}) is the smallest n∈ωn\in\omega, so that for all A⊆NA\subseteq \mathbb{N} of size n+1, the orbit of every a∈Aa\in A under the pointwise stabilizer of A∖{a}A\setminus\{a\} is finite. We study the Bernoulli shift P↷RNP\curvearrowright \mathbb{R}^{\mathbb{N}} for various Polish permutation groups P and we provide criteria under which the P-shift is generically ergodic relative to the injective part of the Q-shift, when Q has algebraic dimension ≤n\leq n. We use this to show that the sequence of pairwise ∗*-reduction-incomparable equivalence relations defined in [KP21] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal ℵ1+\aleph_1^{+} which strongly resembles the equivalence relation of pinned cardinal ℵ1+\aleph_1^{+} from [Zap11], but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group Q. We show that when Q is "locally finite" -- e.g. when Q=Aut(M)Q=\mathrm{Aut}(\mathcal{M}), where M\mathcal{M} is a locally finite countable structure with no algebraicity -- the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.
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... Proof of Application 3. The implication from right to left is immediate by Theorem 4.1 and [Ke92]. For the left-to-right implication, suppose that G is not locally compact and fix an open neighborhood of the identity, V . ...
On Polish groups admitting non-essentially countable actions
Article
Full-text available
  • Dec 2020
It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.
View
... Kechris in [8] proves the following. ...
Minimal model-universal flows for locally compact Polish groups
Preprint
  • Jun 2020
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.
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... Following [12] we say that E X G is (C) σ-lacunary if there are sequences of Borel sets {B n } n<ω and {V n } n<ω such that n<ω B n is a countable complete section of E X G , V n ⊆ G is an open neighbourhood of 1 G and B n is V n -lacunary for every n ∈ N, i.e., if g · x = y for some g ∈ V n and x, y ∈ B n , then x = y. It follows from [9] that in the case when G is a locally compact Polish group, then (A) and (C) are equivalent. Main result of this paper is the following statement. ...
\sigma$-Lacunary actions of Polish groups
Preprint
  • Jan 2020
We show that every essentially countable orbit equivalence relation induced by a continuous action of a Polish group on a Polish space is σ\sigma-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.
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... • ( [Kec92]) If G is a Polish locally compact group and X is a Borel G-space, then the orbit equivalence relation of G X admits classification by countable structures. ...
Non-classification of Cartan subalgebras for a class of von Neumann algebras
Article
  • Oct 2017
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II1_1 factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II1_1 factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II1_1 factor up to unitary conjugacy are not classifiable by countable structures.
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... Indeed a neighborhood basis at the identity consists of the pointwise stabilizers of finite sets, and it is easy to see from the arguments of Theorem 2.1 that such stabilizers are compact. We may now appeal to the well-known result from [Kec92] which implies that any orbit equivalence relation induced by a continuous action of a locally compact group is Borel reducible to E ∞ . ...
On the classification of automorphisms of trees
Article
  • Sep 2017
We identify the complexity of the classification problem for automorphisms of a given countable regularly branching tree up to conjugacy. We consider both the rooted and unrooted cases. Additionally, we calculate the complexity of the conjugacy problem in the case of automorphisms of several non-regularly branching trees.
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... We say that a Borel subset Σ Ă Z is a discrete section of the action of R if, for any z P Z, the set of visit times tr P R : rz P Σu is discrete and closed in R. The main theorem of [12] shows that there is a discrete section Σ for the action of R such that RΣ " Z. ...
Translation of the paper `Mesures stationnaires et ferm'es invariants des espaces homog'enes I', by Y. Benoist and J.-F. Quint
Article
  • Oct 2016
A translation of the famous paper of Benoist and Quint, `Mesures stationnaires et ferm\'es invariants des espaces homog\'enes I', Ann. Math. 174 (2011).
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... According to a theorem of Kechris [4], we may find a Borel subset Σ of X such that 1. Any R-orbit in X meets Σ, i.e. ...
Projections of Patterson-Sullivan measures and the Oh-Mohammadi dichotomy
Article
  • May 2016
Let Γ\Gamma be some discrete subgroup of SOo(n+1,R)\mathbf{SO}^o(n+1,\mathbf{R}) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SOo(n+1,R)=KAN\mathbf{SO}^o(n+1,\mathbf{R})=KAN. We also study the dimension of projected Patterson-Sullivan measures along some fixed small circle.
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... While the notion of a cross section makes sense for actions of any Polish group, its simple geometric interpretation as in Figure 3 seems to be specific to actions of the real line. In the language of cross sections Theorem 2.1 is valid for all locally compact groups as showed by Kechris in [Kec92]. ...
Regular cross sections of Borel flows
Article
  • Jul 2015
Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.
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Representation of measurable multidimensional flows by Lipschitz functions
Preprint
Full-text available
  • May 2025
We show that the shift space of 1-Lipschitz functions from Rk\mathbb{R}^k to the unit interval is a topological model for all free measurable Rk\mathbb{R}^k-flows. This generalizes a theorem from 1973 by Eberlein for R\mathbb{R}-flows. For k≥2k\geq 2 the proof relies on an Rk\mathbb{R}^k-generalization of a Lipschitz refinement of the Bebutov--Kakutani theorem proven by Gutman, Jin and Tsukamoto. A separate expository proof of Eberlein's theorem (k=1) is included (differing from the original proof).
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Siegel-Radon transforms of transverse dynamical systems
Preprint
Full-text available
  • May 2025
We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group G, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over G together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or Θ\Theta-) transforms and Marklof-Str\"ombergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis. Our main applications concern dynamical systems (X,μ)(X, \mu) in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced G-representations into Lp(X,μ)L^p(X, \mu) for appropriate values of p. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case. In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schr\"odinger representations into the L2L^2-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms.
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The classification problem for operator algebraic varieties and their multiplier algebras
Preprint
  • Aug 2015
We study from the perspective of Borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties. These algebras are precisely the multiplier algebras of irreducible complete Nevanlinna-Pick spaces. We prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants. In order to prove such a result, we develop the theory of turbulence for Polish groupoids, which generalizes Hjorth's turbulence theory for Polish group actions. We also prove that the classification problem for multiplier algebras associated with varieties in a finite dimensional ball up to isometric isomorphism has maximum complexity among the essentially countable classification problems. In particular, this shows that Blaschke sequences are not smoothly classifiable up to conformal equivalence via automorphisms of the disc.
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Katok’s special representation theorem for multidimensional Borel flows
Article
Full-text available
  • Oct 2023
Katok’s special representation theorem states that any free ergodic measure- preserving Rd\mathbb {R}^{d} -flow can be realized as a special flow over a Zd\mathbb {Z}^{d} -action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel Rd\mathbb {R}^{d} -flows emerge from Zd\mathbb {Z}^{d} -actions through the special flow construction using bi-Lipschitz cocycles.
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CONTINUOUS LOGIC AND BOREL EQUIVALENCE RELATIONS
Article
  • Jun 2022
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 Σ20\mathbf {\Sigma }^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 Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.
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Abelian group actions and hypersmooth equivalence relations
Article
  • Mar 2022
  • ANN PURE APPL LOGIC
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.
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Continuous logic and Borel equivalence relations
Preprint
  • Sep 2021
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 Σ20\mathbf{\Sigma}^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 Kechris's theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.
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A dichotomy for countable unions of smooth Borel equivalence relations
Preprint
  • May 2021
We show that if an equivalence relation E on a Polish space is a countable union of smooth Borel subequivalence relations, then there is either a Borel reduction of E to a countable Borel equivalence relation on a Polish space or a continuous embedding of E1\mathbb{E}_1 into E. We also establish related results concerning countable unions of more general Borel equivalence relations.
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The Feldman-Moore, Glimm-Effros, and Lusin-Novikov theorems over quotients
Preprint
  • May 2021
We establish generalizations of the Feldman-Moore theorem, the Glimm-Effros dichotomy, and the Lusin-Novikov uniformization theorem from Polish spaces to their quotients by Borel orbit equivalence relations.
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One-ended spanning subforests and treeability of groups
Preprint
  • Apr 2021
We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests.
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RECURRENCE AND THE EXISTENCE OF INVARIANT MEASURES
Article
  • Feb 2021
We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.
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The classification problem for torsion-free abelian groups of finite rank
Article
  • Oct 2002
We prove that for each n ≥ 1 n \geq 1 , the classification problem for torsion-free abelian groups of rank n + 1 n+1 is not Borel reducible to that for torsion-free abelian groups of rank n n .
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Countable sections for locally compact group actions. II
Article
  • Jan 1994
In this paper we study the structure of the orbit equivalence relation induced by a Borel action of a second countable locally compact group on a standard Borel space.
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The classification problem for operator algebraic varieties and their multiplier algebras
Article
  • Aug 2015
We study from the perspective of Borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties. These algebras are precisely the multiplier algebras of irreducible complete Nevanlinna-Pick spaces. We prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants. In order to prove such a result, we develop the theory of turbulence for Polish groupoids, which generalizes Hjorth's turbulence theory for Polish group actions. We also prove that the classification problem for multiplier algebras associated with varieties in a finite dimensional ball up to isometric isomorphism has maximum complexity among the essentially countable classification problems. In particular, this shows that Blaschke sequences are not smoothly classifiable up to conformal equivalence via automorphisms of the disc.
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Lebesgue Orbit Equivalence of Multidimensional Borel Flows
Article
  • Apr 2015
The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue Orbit Equivalence, by which we understand an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non smooth Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they admit the same number of invariant ergodic probability measures.
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The classification problem for finite rank Butler groups
Article
  • Dec 2008
We consider the complexity of the isomorphism and quasi-isomorphism problems for finite rank Butler groups, as well as the related question of the complexity of the classification problem for representations of finite posets over various fields.
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Incomparable treeable equivalence relations
Article
  • Jul 2012
We establish Hjorth’s theorem that there is a family of continuum-many pairwise strongly incomparable free actions of free groups, and therefore a family of continuum-many pairwise incomparable treeable equivalence relations.
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On the Measurability of Orbits in Borel Actions
Article
Full-text available
  • Mar 1977
We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.
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A selection theorem for group actions
Article
Full-text available
  • Feb 1979
  • PAC J MATH
Let a Polish group G act continuously on a Polish space X, inducing an equivalence relation E. Let Ey be the restriction of E to an invariant Borel subset Y of X. Assume Ey is countably separated. Then it has a Borel transversal.
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A Glimm-Effros dichotomy for Borel equivalence relations
Article
  • Jan 1969
  • J AM MATH SOC
A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in the Polish group action case when additionally the induced equivalence relation is Fσ. It is the purpose of this paper to extend the Glimm-Effros dichotomy to the very general context of an arbitrary Borel equivalence relation (not even necessarily induced by a group action). Despite the totally classical descriptive set-theoretic nature of our result, our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability (or recursion) theory on the integers.
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Ergodic equivalence relations, cohomology, and von Neumann algebras. I
Article
  • Jan 1977
Let (X, B) be a standard Borel space, R Ì X x X an equivalence. relation £& x Assume each equivalence class is countable. Theorem 1:3 a countable group G of Borel isomorphisms of (X, $) so that R - {(*, gx): g Î G. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]—[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of “module over R” is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let a, p be rationally independent irrationals on the circle T, and /Borel: T->T. Then 3 Borel g, h: T-*T with f(x) - (g(ax)/g(x))(h(bx)/h(x)) a.e. The notion of “skew product action*' is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of c, defined in terms of the skew action. See also Schmidt [1].
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Orbit structure and countable sections for actions of continuous groups
Article
  • Jun 1978
It is shown that if a second countable locally compact group G acts nonsingularly on an analytic measure space (S, μ), then there is a Borel subset E ⊂ S such that EG is conull in S and each sG ∩ E is countable. It follows that the measure groupoid constructed from the equivalence relation s ∼ sg on E may be simply described in terms of the measure groupoid made from the action of some countable group. Some simplifications are made in Mackey's theory of measure groupoids. A natural notion of “approximate finiteness” (AF) is introduced for nonsingular actions of G, and results are developed parallel to those for countable groups; several classes of examples arising naturally are shown to be AF. Results on “skew product” group actions are obtained, generalizing the countable case, and partially answering a question of Mackey. We also show that a group-measure space factor obtained from a continuous group action is isomorphic (as a von Neumann algebra) to one obtained from a discrete group action.
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Measure and category in effective descriptive set theory
Article
  • Jul 1973
  • Ann Math Logic
We are concerned in this paper with some definability aspects of the theory of measure and category on the continuum. Our objects of study are the projective subsets of the reals and their structure from a measure theoretic and topological point of view.
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Local product structure for group actions
Article
  • Mar 1991
  • ERGOD THEOR DYN SYST
A differentiable G-space is introduced, for a Lie group G, into which every countably separated Borel G-space can be imbedded. The imbedding can be a continuous map if the space is a separable metric space. Such a G-space is called a universal G-space. This universal G-space has a local product structure for the action of G. That structure is inherited by invariant subspaces, giving a local product structure on general G-spaces. This information is used to prove that G-spaces are stratified by the subsets consisting of points whose orbits have the same dimension, to prove that G-spaces with stabilizers of constant dimension are foliated, to give a short proof that closed subgroups of Lie groups are Lie groups, to give a new proof and a stronger version of the Ambrose—Kakutani Theorem and to give a new proof of the existence of near-slices at points having compact stabilizers and hence of the existence of slices for Cartan G-spaces.
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A descriptive version of Ambrose’s representation theorem for flows
Article
  • Dec 1988
We prove here an analogue of Ambrose-Kakutani representation theorem for measurable flows. No measure is required and no points are dropped. This helps us to generalize a theorem due to Shelah and Weiss and answer a question due to A Ramsay.
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Topologies on measured groupoids
Article
  • Jul 1982
If an analytic Borel group G has a quasiinvariant measure, it is known that G is actually a locally compact group with the original Borel structure being generated by the topology and the original measure being equivalent to Haar measure. In this paper a variation is given on the known proof which then extends to show that an analytic measured groupoid has a σ-compact, and also a locally compact, inessential reduction which is a topological groupoid. In the σ-compact case, it is proved that every “almost” homomorphism agrees a.e. with a (strict) homomorphism. Also, the topology is used to show that every measured groupoid has a complete countable section ¦7¦ and that every locally compact equivalence relation has a complete transversal ¦3¦. These are further used to show that some results of Feldman et al. ¦7¦ apply in general and that a locally compact groupoid with (continuous) Haar system has sufficiently many non-singular Borel G-sets provided that the orbit measures are atom-free ¦23¦.
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Virtual subgroups of ℝ″ and ℤ″
  • Jan 1974
  • 187
  • Forrest