Ends of Schreier graphs of hyperbolic groups

TL;DR

This paper investigates the ends of Schreier graphs associated with hyperbolic groups, providing an algorithm for the case where the subgroup is quasi-convex, addressing a previously uncomputable problem.

Contribution

It introduces an algorithm to determine the number of ends of Schreier graphs for hyperbolic groups with quasi-convex subgroups, a problem previously deemed uncomputable.

Findings

Algorithm successfully computes the number of ends for quasi-convex subgroups

Provides new insights into the structure of Schreier graphs of hyperbolic groups

Advances understanding of subgroup properties in hyperbolic groups

Abstract

We study the number of ends of a Schreier graph of a hyperbolic group. Let G be a hyperbolic group and let H be a subgroup of G. In general, there is no algorithm to compute the number of ends of a Schreier graph of the pair (G, H). However, assuming that H is a quasi-convex subgroup of G, we construct an algorithm.