The Martin Boundary of Relatively Hyperbolic Groups with Virtually Abelian Parabolic Subgroups

Abstract

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space ${\mathcal H}^n$, we show that the Martin boundary coincides with the $CAT(0)$ boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.