Related papers: Expansive algebraic actions of discrete residually…

For a countable amenable group \Gamma and an element f in the integral group ring Z\Gamma being invertible in the group von Neumann algebra of \Gamma, we show that the entropy of the shift action of \Gamma on the Pontryagin dual of the…

We survey some of the known criteria for expansiveness of principal algebraic actions of countably infinite discrete groups. In the special case of the discrete Heisenberg group we propose a new approach to this problem based on Allan's…

We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.

Using periodic points we study a notion of entropy with values in the p-adic numbers. This is done for actions of countable discrete residually finite groups $\Gamma$. For suitable $\Gamma = \mathbb{Z}^d$-actions we obtain p-adic analogues…

We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented…

An algebraic $\Gamma$-action is an action of a countable group $\Gamma$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $\Gamma$ on a…

An \textit{algebraic} action of a discrete group $\Gamma $ is a homomorphism from $\Gamma $ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $\Gamma $ is determined by a module…

This paper gives an algebraic characterization of expansive actions of countable abelian groups on compact abelian groups. This naturally extends the classification of expansive algebraic $\mathbb{Z}^d$-actions given by Schmidt using…

In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary…

Consider a lattice $\Gamma$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $\Gamma$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its…

We prove that any finitely generated elementary amenable group of zero (algebraic) entropy contains a nilpotent subgroup of finite index or, equivalently, any finitely generated elementary amenable group of exponential growth is of…

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, and $\Gamma$ a subgroup of the affine group Aff$(G)$ stabilizing $\Lambda$. Then the action of $\Gamma$ on…

We introduce a notion of entropy for automorphisms of discrete groups which admit amenable actions on a compact space. This entropy is dual to classical topological entropy in the sense that if G is discrete and abelian then our notion of…

Let $\Gamma$ be a countable group. A $\Gamma$-action on a compact abelian group $X$ by continuous automorphisms of $X$ is called Noetherian if the dual of $X$ is Noetherian as a ${\mathbb Z}(\Gamma)$-module. We prove that any Noetherian…

For every finite-to-one map $\lambda:\Gamma\to\Gamma$ and for every abelian group $K$, the generalized shift $\sigma_\lambda$ of the direct sum $\bigoplus_\Gamma K$ is the endomorphism defined by…

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…

Given a finitely generated group $\Gamma$, we study the space ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$ of all actions of $\Gamma$ by isometries of the rational Urysohn metric space ${\mathbb Q\mathbb U}$, where ${\rm Isom}(\Gamma,{\mathbb…

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq \alpha\leq\beta\leq1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite…

Fixing an arithmetic lattice $\Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ with $[\Gamma : \Gamma \cap \Delta] [\Delta: \Gamma \cap \Delta] =…

Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on F\o lner sequences for amenable groups, entropy defined using a random ordering of…