Related papers: Bernoulli actions are weakly contained in any free…

Answering Vershik's question we show that quasi-similarity does not conserve the entropy, proving quasi-similarity of all Bernoulli actions of a countable infinite group. We prove also the following generalization of Pinsker's theorem: the…

We survey recent progress in the theory of weak containment of measure preserving group actions.

Let $\eta$ be an arbitrary countable ordinal. Using results of Bourgain and Gamburd on compact systems with spectral gap we show the existence of an action of the free group on three generators $F_3$ on a compact metric space $X$, admitting…

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the…

Let $G$ be a discrete countable infinite group that does not have Kazhdan's property ~(T) and let $\kappa$ be a generating probability measure on $G$. Then for each $t>0$, there is a type $III_1$ ergodic free nonsingular $G$-action whose…

We give an elementary proof for Lewis Bowen's theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K and…

We study probability measure preserving (p.m.p.) non-free actions of free groups and the associated IRS's. The perfect kernel of a countable group Gamma is the largest closed subspace of the space of subgroups of Gamma without isolated…

We introduce computable actions of computable groups and prove the following versions of effective Birkhoff's ergodic theorem. Let $\Gamma$ be a computable amenable group, then there always exists a canonically computable tempered two-sided…

A weakly continuous near-action of a Polish group $G$ on a standard Lebesgue measure space $(X,\mu)$ is whirly if for every $A\subseteq X$ of strictly positive measure and every neighbourhood $V$ of identity in $G$ the set $VA$ has full…

Dan Rudolph showed that for an amenable group $\Gamma$, the generic measure-preserving action of $\Gamma$ on a Lebesgue space has zero entropy. Here this is extended to nonamenable groups. In fact, the proof shows that every action is a…

We show that if $\Gamma\curvearrowright (X^\Gamma,\mu^\Gamma)$ is a Bernoulli action of an i.c.c. nonamenable group $\Gamma$ which is weakly amenable with Cowling-Haagerup constant $1$, and $\Lambda\curvearrowright(Y,\nu)$ is a free ergodic…

We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show…

We prove that if $G$ is a finitely generated group and $Z$ is a uniformly recurrent subgroup of $G$ then there exists a minimal system $(X,G)$ with $Z$ as its stability system. This answers a query of Glasner and Weiss \cite{GW} in the case…

We prove that for an arbitrary indexing group, every ergodic infinitely divisible stationary process that is separable in probability is weakly mixing. This shows that, as in the well-known case of Gaussian stationary processes, ergodicity…

We study a notion of entropy for probability measure preserving actions of finitely generated free groups, called f-invariant entropy, introduced by Lewis Bowen. In the degenerate case, the f-invariant entropy is negative infinity. In this…

We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(\Lambda^G,\sigma)$, a typical measure has entropy $c$ and is Bernoulli.…

We say that two free probability-measure-preserving actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the…

Let $m\in\mathbb{N}$ and $\textbf{X}=(X,\mathcal{X},\mu,(T_{\alpha})_{\alpha\in\mathbb{R}^{m}})$ be a measure preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\nu$ on $\mathbb{R}^{m}$ is weakly equidistributed…

We prove W$^*$-superrigidity for a large class of coinduced actions. We prove that if $\Sigma$ is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group $\Gamma$, the coinduced action…

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group $G$, a word $w$ in the free group on $k$ generators induces…