Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property

Abstract

Let μ and ν be probability measures on a group Γ and let Gμ and Gν denote Green’s function with respect to μ and ν. The group Γ is said to admit instability of Green’s function if there are symmetric, finitely supported measures μ and ν and a sequence {xn} such that Gμ(e, xn)/Gν(e,xn) →0, and Γ admits instability of recurrence if there is a set S that is recurrent with respect to ν but transient with respect to μ. We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously known groups with these instabilities did not have the Liouville property.