On a theorem of Avez

Abstract

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