The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups

TL;DR

This paper characterizes the Martin boundary for random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups, revealing topological structures including the Sierpinski carpet in specific cases.

Contribution

It provides a complete topological description of the Martin boundary for these groups, linking it to CAT(0) boundaries and geometric structures.

Findings

Martin boundary coincides with the CAT(0) boundary in hyperbolic lattice cases

In dimension 3, the boundary is homeomorphic to the Sierpinski carpet

Provides a topological classification for the Martin boundary in these groups

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 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.