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Let $A,B\subset M$ be inclusions of $\sigma$-finite von Neumann algebras such that $A$ and $B$ are images of faithful normal conditional expectations. In this article, we investigate Popa's intertwining condition $A\preceq_MB$ using their modular actions. In the main theorem, we prove that if $A\preceq_MB$ holds, then an intertwining element for $A\preceq_MB$ also intertwines some modular flows of $A$ and $B$. As a result, we deduce a new characterization of $A\preceq_MB$ in terms of their continuous cores. Using this new characterization, we prove the first W$^*$-superrigidity type result for group actions on amenable factors. As another application, we characterize stable strong solidity for free product factors in terms of their free product components., Comment: 35 pages

We say that two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits. In particular the Ornstein-Weiss theorem implies that all infinite amenable groups are orbit equivalent to the group of integers. To refine this notion Delabie, Koivisto, Le Ma\^itre and Tessera introduced a quantitative version of orbit equivalence and of its measure theoretic counterpart called measure equivalence. They furthermore obtained an upper bound to the possible quantification between two given groups. In this article we offer to answer the inverse problem (find a group being orbit or measure equivalent to a prescribed group with prescribed quantification) in the case of the group of integers, the lamplighter group and Brieussel-Zheng's diagonal products. These results moreover show that the upper bound given by Delabie et al. is optimal: the orbit and measure equivalences we obtain all realize the aforementioned upper bound., Comment: 59 pages, 20 figures

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., 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 give a sufficient condition for parameter rigidity of actions of solvable Lie groups, by vanishing of (uncountably many) first cohomologies of the orbit foliations. In some cases, we can prove that vanishing of finitely many cohomologies is sufficient. For this purpose we use a rigidity property of quasiisometry. As an application we prove some actions of 2-step solvable Lie groups on mapping tori are parameter rigid. Special cases of these actions are considered in a paper of Matsumoto and Mitsumatsu. We also remark on the relation between transitive locally free actions of solvable Lie groups and lattices in solvable Lie groups, and apply results in rigidity theory of lattices in solvable Lie groups to construct transitive locally free actions with some properties., Comment: 35 pages, no figures

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph $\cG$ is uniformly hyperfinite if for any $\eps>0$ there exists $K\geq 1$ such that not only $\cG$, but all of its subgraphs of positive measure are $(\eps,K)$-hyperfinite. We also show that this condition is equivalent to weighted hyperfiniteness and a strong version of fractional hyperfiniteness, a notion recently introduced by Lov��sz. As a corollary, we obtain a characterization of exactness of finitely generated groups via uniform hyperfiniteness., 21 pages

In this paper, we study holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is the solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the noncommutative case nonlinearizable semicocycles are shown to exist. We derive simple conditions for linearizability and show that they are sharp.

We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a "conditional" Pontryagin duality for spaces of abstract measurable maps., Comment: 28 pages [v4]: removed material concerning weak conditional elements, following an error pointed out by the referee. Also modified the notation in line with referee comments on this paper and related papers

Supplementary results obtained after the completion of our previous paper are given together with discussing some examples. A quick review of the previous paper is also included.

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

We prove the following indistinguishability theorem for $k$-tuples of trees in the uniform spanning forest of $\mathbb{Z}^d$: Suppose that $\mathscr{A}$ is a property of a $k$-tuple of components that is stable under finite modifications of the forest. Then either every $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely, or no $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liouville property and for which every component of the USF is one-ended., Comment: 14 pages. This theorem and its proof appeared in the first version of arXiv:1702.05780

We present a general setting to investigate U_fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some U_fin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be U_fin-cocycle superrigid., Comment: Generalized Theorem 3.1 and added Theorem 5.3.

Consider an invertible measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the later papers of del Junco-Reinhold-Weiss and del Junco-Begun we obtain Krengel- type results for weakly mixing random dynamical systems (or equivalently, skew products that are relatively weakly mixing).

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

We give conditions for the exactness of Rokhlin endomorphisms, apply these to random walks on locally compact, second countable topological groups and obtain that the action on the Poisson boundary of an adapted random walk on such a group is weakly mixing.

We give an explicit description of all finite Borel measures on Noetherian topological spaces X, and characterize them as objects dual to a space of functions on X. We use these results to study the asymptotic behavior of continuous dynamical systems on Noetherian spaces., Comment: Minor revisions, results remain the same. To appear in the Journal of Geometric Analysis

To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of Baumslag-Solitar groups under weak orbit equivalence. Results on groups which are measure equivalent to Baumslag-Solitar groups are also provided., Comment: 49 pages, minor corrections

We prove that the group-measure-space von Neumann algebra $L^\infty(T^2) \rtimes SL(2,Z)$ is solid. The proof uses topological amenability of the action of $SL(2,Z)$ on the Higson corona of $Z^2$., 4 pages

Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if $G_1,G_2$ are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over $G_1$ is stably orbit equivalent to every nontrivial Bernoulli shift over $G_2$. This answers a question of S. Popa., Comment: The second version has minor notational changes

We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of L��ck, Sauer and Wegner hold for treeable equivalence relations., 18 pages

We define lines of minima in the thick part of Outer space for the free group Fn with n>2 generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of Fn defines such a line a minima. Now let G be a subgroup of the outer automorphism group of Fn which is not virtually abelian. We obtain as an immediate application that if G contains at least one fully irreducible element then for every p, Final version. 35 p

This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors., Comment: 24 pages, continuation of arXiv:0706.3623

Using an aperiodic and repetitive subtree of the Cayley graph of the free Abelian group with two generators, described by Kenyon, Ghys has constructed an example of minimal Riemann surface lamination having both Euclidean and hyperbolic leaves. We prove that the transverse dynamics of this lamination is represented (in a measurable way) by a 2-adic odometer. In fact, we can describe its topological transverse dynamics, and prove that the Ghys-Kenyon lamination is affable., Comment: 16 pages, 6 figures, to appear in Asterisque

We show that the "twisted" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay., Comment: 11 pages

We show that a sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous pieces. The result can be viewed as a "finitarization" of the classical Farrell-Varadarajan Ergodic Decomposition Theorem., Corrected an error in the proof of the Homogeneity Lemma. A new result is proved about edge-colorings of convergent graph sequences

We prove that if $G$ is a countable discrete group with property (T) over an infinite subgroup $H

We study the mapping class groups of compact orientable surfaces from the viewpoint of measure equivalence. In this paper, we announce some classification result of the mapping class groups in terms of measure equivalence and the result that there exist various kinds of discrete groups which are not measure equivalent to the mapping class groups. As a byproduct of the proof, it turns out that the mapping class group is exact.

These notes contain an Ergodic-theoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of ``rigid'' groups) satisfy the assumptions of Popa's theorem., Comment: v.2: slight changes in the presentation, (some) typos corrected, wq-normal subgroups added. 35 pages

We prove that solutions $u$ of the equation $f=u-u\circ T$ are automatically Holder continuous when $f$ is Holder continuous, and $T$ is non uniformly expanding and Markov. This result applies in particular to Young towers and to intermittent maps.

There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu=f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.

If $M$ is a (separable) von Neumann algebra and $A$ is a Cartan subalgebra of $M$ , then $M$ is determined by an equivalence relation and a 2-cocycle. By constructing an equivalence subrelation, we show that for any intermediate von Neumann subalgebra $N$ between $M$ and $A$ , there exists a faithful normal conditional expectation from $M$ onto $N$ .

We define an analog of Voiculescu's free entropy for $n$-tuples of unitaries $u\sb 1,\ldots u\sb n$ in a tracial von Neumann algebra $M$ which normalize a unital subalgebra $L\sp \infty[0,1]=B\subset M$. Using this quantity, we define the free dimension $\delta\sb 0(u\sb 1,\ldots,u\sb n\between B)$. This number depends on $u\sb 1,\ldots u\sb n$ only up to orbit equivalence over $B$. In particular, if $R$ is a measurable equivalence relation on $[0,1]$ generated by $n$ automorphisms $\alpha\sb 1,\ldots \alpha\sb n$, let $u\sb 1,\ldots u\sb n$ be the unitaries implementing $\alpha\sb 1,\ldots \alpha\sb n$ in the Feldman-Moore crossed product algebra $M=W\sp \ast([0,1],R)\supset B=L\sp \infty[0,1]$. Then the number $\delta(R)=\delta\sb 0(u\sb 1,\ldots u\sb n\between B)$ is an invariant of the equivalence relation $R$. If $R$ is treeable, $\delta(R)$ coincides with the cost $C(R)$ of $R$ in the sense of D. Gaboriau. In particular, it is $n$ for an equivalence relation induced by a free action of the free group $\mathbb {F}\sb n$. For a general equivalence relation $R$ possessing a finite graphing of finite cost, $\delta(R)\leq C(R)$. Using the notion of free dimension, we define a dynamical entropy invariant for an automorphism of a measurable equivalence relation (or, more generally, of an $r$-discrete measure groupoid) and give examples.

We extend the Delorme-Guichardet characterization of Kazhdan property $T$ groups to $r$-discrete measured groupoids. We give several applications, in particular to stability results of Kazhdan property $T$ and to the study of cocycles taking values in a group having the Haagerup property., Comment: 43 pages

We yield operators $U$ and $V$ on Hilbert spaces that are parameterized by the orbits of certain interval maps that exhibit chaotic behavior and obey the (deformed) Baumslag--Solitar relation $$UV=e^{2\pi i \alpha} VU^n,\qquad \alpha\in \mathbb{R},\ n\in\mathbb{N}.$$ We then prove that the scalar $e^{2\pi i \alpha}$ can be removed whilst retaining the isomorphism class of the $C^*$-algebra generated by $U$ and $V$. Finally, we simultaneously unitarize $U$ and $V$ by gluing pairs of orbits of the underlying noninvertible dynamical system and investigate these unitary representations under distinct pairs of orbits.