Ergodicity of principal algebraic group actions

An \textit{algebraic} action of a discrete group $Γ$ is a homomorphism from $Γ$ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $Γ$ is determined by a module $M=\widehat{X}$ over the integer group ring $\mathbb{Z}Γ$ of $Γ$. The simplest examples of such modules are of the form $M=\mathbb{Z}Γ/\mathbb{Z}Γf$ with $f\in \mathbb{Z}Γ$; the corresponding algebraic action is the \textit{principal algebraic $Γ$-action} $α_f$ defined by $f$. In this note we prove the following extensions of results by Hayes \cite{Hayes} on ergodicity of principal algebraic actions: If $Γ$ is a countably infinite discrete group which is not virtually cyclic, and if $f\in\mathbb{Z}Γ$ satisfies that right multiplication by $f$ on $\ell ^2(Γ,\mathbb{R})$ is injective, then the principal $Γ$-action $α_f$ is ergodic (Theorem \ref{t:ergodic2}). If $Γ$ 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 $\mathbb{Z}Γ$ (Theorem \ref{t:ergodic1}). Finally, if $Γ$ is torsion-free, not virtually cyclic, and satisfies Linnell's \textit{analytic zero-divisor conjecture}, then $α_f$ is ergodic for every $f\in \mathbb{Z}Γ$ (Remark \ref{r:analytic zero divisor}).

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