Related papers: Inner amenability, property Gamma, McDuff II_1 fac…

We provide a fairly large class of II$_1$ factors $N$ such that $M=N\bar{\otimes}R$ has a unique McDuff decomposition, up to isomorphism, where $R$ denotes the hyperfinite II$_1$ factor. This class includes all II$_1$ factors…

Given a topologically free action of a countably infinite amenable group on the Cantor set, we prove that, for every subgroup $G$ of the topological full group containing the alternating group, the group von Neumann algebra $\mathscr{L} G$…

We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain…

Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the…

A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type II_1 under direct product. We show that for a countable group G and…

Let $G$ be a subgroup of a discrete (countable) group $\Gamma$. We introduce a notion of relative inner amenability of $G$ in $\Gamma$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss…

We give a simple and explicit proof that the free group $\mathbb{F}_2$ admits a measure equivalence embedding into any nonamenable locally compact second countable (lcsc) group $G$. We use this to prove that every nonamenable lcsc group $G$…

Let us say that a discrete countable group is stable if it has an ergodic, free, probability-measure-preserving and stable action. Let G be a discrete countable group with a central subgroup C. We present a sufficient condition and a…

We study hereditary properties of the class of countable groups admitting an amenable, transitive and faithful action on a countable set. We consider mainly the case of amalgamated free products, and we show in particular that the double of…

We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin, our result recovers that mapping class groups and Out(F_n) are not…

A measure-preserving action of a discrete countable group on a standard probability space is called stable if the associated equivalence relation is isomorphic to its direct product with the ergodic hyperfinite equivalence relation of type…

It is well known that if $G$ is a countable amenable group and $G \curvearrowright (Y, \nu)$ factors onto $G \curvearrowright (X, \mu)$, then the entropy of the first action must be greater than or equal to the entropy of the second action.…

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…

Let $\alpha: G\curvearrowright X$ be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. In this paper, under the hypothesis that the invariant ergodic probability Borel measure…

Let a countable amenable group $G$ act on a \zd\ compact metric space $X$. For two clopen subsets $\mathsf A$ and $\mathsf B$ of $X$ we say that $\mathsf A$ is \emph{subequivalent} to $\mathsf B$ (we write $\mathsf A\preccurlyeq \mathsf…

We prove pointwise and maximal ergodic theorems for probability measure preserving (p.m.p.) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type $III_1$. We show that this…

Let $\Gamma$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $\Gamma$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the…

We study stability properties of amenable locally compact quantum groups under the bicrossed product construction. We obtain as our main result an equivalence between amenability of the bicrossed product and amenability of the matched…

Let $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss the corresponding…

Let $M$ be a finite von Neumann algebra with the Haagerup property, and let $G$ be a compact group that acts continuously on $M$ and that preserves some finite trace $\tau$. We prove that if $\Gamma$ is a countable subgroup of $G$ which has…