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 Γand let G_μand G_νdenote Green's function with respect to μand ν. The group Γis said to admit instability of Green's function if there are symmetric, finitely supported measures $μ$ and νand a sequence \{x_n\} such that G_μ(e, x_n)/G_ν(e,x_n) \to 0, and Γadmits instability of recurrence if there is a set S that is recurrent with respect to νbut 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.