Ergodicity of principal algebraic group actions

Abstract

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 $M=\widehat{X}$ over the integer group ring $\mathbb{Z}\Gamma $ of $\Gamma $. The simplest examples of such modules are of the form $M=\mathbb{Z}\Gamma /\mathbb{Z}\Gamma f$ with $f\in \mathbb{Z}\Gamma $; the corresponding algebraic action is the \textit{principal algebraic $\Gamma $-action} $\alpha _f$ defined by $f$. In this note we prove the following extensions of results by Hayes \cite{Hayes} on ergodicity of principal algebraic actions: If $\Gamma $ is a countably infinite discrete group which is not virtually cyclic, and if $f\in\mathbb{Z}\Gamma $ satisfies that right multiplication by $f$ on $\ell ^2(\Gamma ,\mathbb{R})$ is injective, then the principal $\Gamma $-action $\alpha _f$ is ergodic (Theorem \ref{t:ergodic2}). If $\Gamma $ contains a finitely generated subgroup with a single end (e.g. a finitely generated amenable subgroup which is not virtually cyclic), or an infinite nonamenable subgroup with vanishing first $\ell ^2$-Betti number (e.g., an infinite property $T$ subgroup), the injectivity condition on $f$ can be replaced by the weaker hypothesis that $f$ is not a right zero-divisor in $\mathbb{Z}\Gamma $ (Theorem \ref{t:ergodic1}). Finally, if $\Gamma $ is torsion-free, not virtually cyclic, and satisfies Linnell's \textit{analytic zero-divisor conjecture}, then $\alpha _f$ is ergodic for every $f\in \mathbb{Z}\Gamma $ (Remark \ref{r:analytic zero divisor}).