The geometry of relative Cayley graphs for subgroups of hyperbolic groups

TL;DR

This paper proves that the relative Cayley graph of a quasiconvex subgroup in a hyperbolic group is Gromov-hyperbolic and shows that simple random walks are transient under certain conditions.

Contribution

It establishes the hyperbolicity of relative Cayley graphs for quasiconvex subgroups and analyzes random walk behavior in this context, extending understanding of geometric group theory.

Findings

Relative Cayley graphs are Gromov-hyperbolic for quasiconvex subgroups.

Random walks on these graphs are transient when G is torsion-free, non-elementary, and H has infinite index.

Provides new insights into the geometry and probabilistic properties of hyperbolic groups and their subgroups.

Abstract

We show that if H is a quasiconvex subgroup of a hyperbolic group G then the relative Cayley graph Y (also known as the Schreier coset graph) for G/H is Gromov-hyperbolic. We also observe that in this situation if G is torsion-free and non-elementary and H has infinite index in G then the simple random walk on Y is transient.