Instability of set recurrence and Green's function on groups with the Liouville property

Abstract

Let $\mu$ and $\nu$ be probability measures on a group \Gamma and let G_\mu and G_\nu denote Green's function with respect to \mu and \nu . The group \Gamma is said to admit instability of Green's function if there are symmetric, finitely supported measures $\mu$ and \nu and a sequence \{x_n\} such that G_\mu(e, x_n)/G_\nu(e,x_n) \to 0, and \Gamma admits instability of recurrence if there is a set S that is recurrent with respect to \nu but transient with respect to \mu . 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.