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Let $\Gamma$ be a countable group that admits an essential measurable splitting (for instance, any group measure equivalent to a free product of nontrivial groups). We show: (1) for any two nontrivial countable groups $B$ and $C$ that are measure equivalent, the wreath product groups $B\wr\Gamma$ and $C\wr\Gamma$ are measure equivalent (in fact, orbit equivalent) -- this is interesting even in the case when the groups $B$ and $C$ are finite; and (2) the groups $B\wr \Gamma$ and $(B\times\mathbf{Z})\wr\Gamma$ are measure equivalent (in fact, orbit equivalent) for every nontrivial countable group $B$. On the other hand, we show that certain wreath product actions are not even stably orbit equivalent if $\Gamma$ is instead assumed to be a sofic icc group that is Bernoulli superrigid, and $B$ and $C$ have different cardinalities., Comment: 37 pages, 1 figure, minor revisions and corrections

We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the F\o lner sequence used to define it. This result has been discovered independently by Sa\'ul Rodr\'iguez Mart\'in. Both ours and Rodr\'iguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $\mathbb{N}$ that do not involve F\o lner sequences remain equivalent for arbitrary countably infinite groups., Comment: Many typos were fixed, the content of the introduction was reorganized, the grammar in some statements of results was corrected, and a short proof was added for a result at the end of Section 4

In this note we study countable subgroups of the full group of a measure preserving equivalence relation. We provide various constraints on the group structure, the nature of the action, and on the measure of fixed point sets, that imply that the subgroup topology is not discrete. We mention various conjectures about discrete subgroups of full groups., Comment: 25 pages

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., Comment: 42 pages, 2 x 2 figures

Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have "fixed priced 1" in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that inner amenable groups have fixed price 1., Comment: Incorporates referee comments. To appear in Proceedings of the AMS

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of groups including the lamplighter group and all virtually solvable groups having finite Pr\"ufer rank. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov's notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero-dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Folner tilings, topological dynamics, and $C^*$-algebras., Comment: Corrected a typo in the proof of Theorem 4.8

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempotent conjugation-invariant means on $G$. We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to "locate" conjugation invariant means on a group $G$ relative to a given normal subgroup $N$ of $G$. We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.

We prove that the alternating group of a topologically free action of a countably infinite group $\Gamma$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $\Gamma$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $\Gamma$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance of measure entropy under bounded orbit equivalence., Comment: 30 pages, to appear in Annales Scientifiques de l'ENS

We show that every countable group with infinite FC-center has the Schmidt property, i.e., admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As its consequence, every countable, inner amenable group with property (T) has the Schmidt property., Comment: 47 pages. The referee's comments incorporated (v3). The appendix by the second author was moved to the last section of the text, and the second author joined as a coauthor. The term "FC-radical" was replaced by "FC-center" with the same meaning (v2)

We introduce inner amenability for discrete p.m.p. groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations which either are stable or have a non-trivial central sequence in their full group., Comment: 58 pages, the referee's comments incorporated and Section 9 added

We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence invariant of every essentially free p.m.p. action of G., Comment: This new version adds more context in the introduction and fixes minor typos

The concept of (stable) weak containment for measure-preserving actions of a countable group $\Gamma$ is analogous to the classical notion of (stable) weak containment of unitary representations. If $\Gamma$ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if $\Gamma$ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when $\Gamma=\mathbb{Z}$ this simplex has only a countable set of extreme points but when $\Gamma$ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when $\Gamma$ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points., Comment: there are only a few minimal changes between this version and the previous

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable., Comment: Minor revisions. Final version

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every $d > 1$ there is a $d$-regular acyclic hyperfinite Borel bipartite graph with no Borel perfect matching. These techniques also give examples of hyperfinite bounded degree Borel graphs for which the Borel local lemma fails, in contrast to the recent results of Cs\'oka, Grabowski, M\'ath\'e, Pikhurko, and Tyros. Related to the Borel Ruziewicz problem, we show there is a continuous paradoxical action of $(\mathbb{Z}/2\mathbb{Z})^{*3}$ on a Polish space that admits a finitely additive invariant Borel probability measure, but admits no countably additive invariant Borel probability measure. In the context of studying ultrafilters on the quotient space of equivalence relations under $\mathrm{AD}$, we also construct an ultrafilter $U$ on the quotient of $E_0$ which has surprising complexity. In particular, Martin's measure is Rudin-Kiesler reducible to $U$. We end with a problem about whether every hyperfinite bounded degree Borel graph has a witness to its hyperfiniteness which is uniformly bounded below in size.

Let $G$ be either a profinite or a connected compact group, and $\Gamma, \Lambda$ be finitely generated dense subgroups. Assuming that the left translation action of $\Gamma$ on $G$ is strongly ergodic, we prove that any cocycle for the left-right translation action of $\Gamma\times\Lambda$ on $G$ with values in a countable group is virtually cohomologous to a group homomorphism. Moreover, we prove that the same holds if $G$ is a (not necessarily compact) connected simple Lie group provided that $\Lambda$ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE - and W$^*$- superrigidity. In particular, we obtain the first examples of compact actions of $\mathbb F_2\times\mathbb F_2$ which are W$^*$-superrigid.

We prove that if a measure distal action $\alpha$ of a countable group $\Gamma$ is weakly contained in a strongly ergodic probability measure preserving action $\beta$ of $\Gamma$, then $\alpha$ is a factor of $\beta$. In particular, this applies when $\alpha$ is a compact action. As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg-Zimmer Structure Theorem., Comment: final version

We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first l2-Betti number of G with that of the stabilizer subgroups. An analogous relationship is also shown to hold for cost. This relationship becomes even more pronounced for transitive amenable actions, leading to a simple criterion for vanishing of the first l2-Betti number and triviality of cost. Moreover, for any marked finitely generated nonamenable group G we establish a uniform isoperimetric threshold for Schreier graphs G/H of G, beyond which the group H is necessarily weakly normal in G. Even more can be said in the particular case of an atomless mean for the conjugation action -- that is, when G is inner amenable. We show that inner amenable groups have cost 1 and moreover they have fixed price. We establish U_{fin}-cocycle superrigidity for the Bernoulli shift of any nonamenable inner amenable group. In addition, we provide a concrete structure theorem for inner amenable linear groups over an arbitrary field. We also completely characterize linear groups which are stable in the sense of Jones and Schmidt. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group H(R), recently studied by Monod, are stable. This includes nonamenable groups constructed by Monod and by Lodha and Moore, as well as Thompson's group F., Comment: Theorem 11 (the cocycle superrigidity statement) has been optimized. The statement is now stronger and simpler, and the proof (in section 5) is shorter. This makes Appendix C obsolete, so it has been removed. The other change is that Remark 7.5 from the old version has been moved to the introduction and is now Theorem 15

We classify the ergodic invariant random subgroups of strictly diagonal limits of finite symmetric groups.

We show that for any infinite countable group $G$ and for any free Borel action $G \curvearrowright X$ there exists an equivariant class-bijective Borel map from $X$ to the free part $\mathrm{Free}(2^G)$ of the $2$-shift $G \curvearrowright 2^G$. This implies that any Borel structurability which holds for the equivalence relation generated by $G \curvearrowright \mathrm{Free}(2^G)$ must hold a fortiori for all equivalence relations coming from free Borel actions of $G$. A related consequence is that the Borel chromatic number of $\mathrm{Free}(2^G)$ is the maximum among Borel chromatic numbers of free actions of $G$. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic $G$-equivariant map to $2^G$ lands in the free part. As a corollary we obtain that for every $\epsilon > 0$, every free pmp action of $G$ has a free factor which admits a $2$-piece generating partition with Shannon entropy less than $\epsilon$. This generalizes a result of Danilenko and Park.

A countable group \Gamma is called shift-minimal if every non-trivial measure preserving action of \Gamma weakly contained in the Bernoulli shift of \Gamma on ([0,1]^\Gamma ,\lambda ^\Gamma) is free. We show that any group \Gamma whose reduced C^*-algebra admits a unique tracial state is shift-minimal, and that any group \Gamma admitting a free measure preserving action of cost>1 contains a finite normal subgroup N such that \Gamma /N is shift-minimal. Any shift-minimal group in turn is shown to have trivial amenable radical. Recurrence arguments are used in studying invariant random subgroups of a wide variety of shift-minimal groups. We also examine continuity properties of cost in the context of infinitely generated groups and equivalence relations. A number of open questions are discussed which concern cost, shift-minimality, C^*-simplicity, and uniqueness of tracial state on C^*_r(\Gamma)., Comment: 55 pages, 1 figure; The section on cost has been largely rewritten and tightened

A measure preserving action of a countably infinite group \Gamma is called totally ergodic if every infinite subgroup of \Gamma acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of \Gamma is totally ergodic then there exists a finite normal subgroup N of \Gamma such that the stabilizer of almost every point is equal to N. Surprisingly the proof relies on the group theoretic fact (proved by Hall and Kulatilaka as well as by Kargapolov) that every infinite locally finite group contains an infinite abelian subgroup, of which all known proofs rely on the Feit-Thompson theorem. As a consequence we deduce a group theoretic characterization of countable groups whose non-trivial Bernoulli factors are all free: these are precisely the groups that posses no finite normal subgroup other than the trivial subgroup.

Ab\'ert-Weiss have shown that the Bernoulli shift s of a countably infinite group \Gamma is weakly contained in any free measure preserving action (mpa) b of \Gamma. We establish a strong version of this result, conjectured by Ioana, by showing that s \times b is weakly equivalent to b. This is generalized to non-free mpa's using random Bernoulli shifts. The result for free mpa's is used to show that isomorphism on the weak equivalence class of a free mpa does not admit classification by countable structures. This provides a negative answer to a question of Ab\'ert and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. An infinite residually finite group \Gamma is said to have EMD if the action p of \Gamma on its profinite completion weakly contains all ergodic mpa's of \Gamma, and \Gamma is said to have property MD if i \times p weakly contains all mpa's of \Gamma, where i denotes the trivial action on a standard non-atomic probability space. Kechris asks if these two properties equivalent and we provide a positive answer by studying the relationship between convexity and weak containment., Comment: 41 pages. This version has minor corrections and updates, including updated references

This note answers a question of Kechris: if $H