On a theorem of Avez

Abstract

For each symmetric, aperiodic probability measure $\mu$ on a finitely generated group $G$, we define a subset $A_{\mu }$ consisting of group elements $g$ for which the limit of the ratio ${\mu }^{\ast n}(g)/{\mu }^{\ast n}(e)$ tends to $1$. We prove that $A_{\mu }$ is a subgroup, is amenable, contains every finite normal subgroup, and $G=A_{\mu }$ if and only if $G$ is amenable. For non-amenable groups we show that $A_{\mu }$ is not always a normal subgroup, and can depend on the measure. We formulate some conjectures relating $A_{\mu }$ to the amenable radical.

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