Related papers: On a theorem Dan Rudolph: Part II: Amenable groups

In the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action. For such an action, we introduce entropy tuples in both topological and measure-theoretic settings and…

Let G be one of the local gauge groups C(X,U(n)), C^\infty(X,U(n)), C(X,SU(n)) or C^\infty(X,SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which…

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is…

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we…

The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture…

Entropy of measure preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are the ones given by the techniques developed by Ollagnier and Pinchon on the one hand and the…

The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\sigma$-finite measure spaces. It is inspired by the very first definition of amenability,…

Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy $c$ are…

Let G be an infinite discrete countable amenable group acting continuously on a Lebesgue space X. In this article, using partition and factor-space, the conditional entropy of the action G is defined. We introduction some properties of…

We undertake a comprehensive study of measure equivalence between general locally compact, second countable groups, providing operator algebraic and ergodic theoretic reformulations, and complete the classification of amenable groups within…

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…

We consider an isomorphism invariant for measure-preserving systems - types of generalized entropy convergence rates. We show the connections of this invariant with the types of Shannon entropy convergence rates. In the case when they…

A probability measure preserving action of a discrete amenable group $G$ is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb{Z}$, an action is dominant if and only if it…

Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of $(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less…

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective…

Let $G$ be a countable infinite discrete amenable group.It should be noted that a $G$-system $(X,G)$ naturally induces a $G$-system $(\mathcal{M}(X),G)$, where $\mathcal{M}(X)$ denotes the space of Borel probability measures on the compact…

Let $G$ be a semisimple Lie group with Haar measure $\mu$ and let $\Gamma$ be an irreducible lattice in $G$. For $g\in G$, we consider left translation $L_g$ acting on $(G\backslash\Gamma,\mu)$. We show that if $L_g$ is $K$ (which is…

Given a countable amenable group G and 0 < L < 1, we give an elementary construction of a type-III:L Bernoulli group action. In the case where G is the integers, we show that our nonsingular Bernoulli shifts have independent and identically…

We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the…

We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to \Pi} (X_0, \mu_0)_g$ for $(X_{0},\mu_{0})$ an arbitrary…