Deciding if a hyperbolic group splits over a given quasiconvex subgroup

TL;DR

This paper introduces an algorithm to determine if a quasiconvex subgroup of a hyperbolic group causes a splitting, and also provides methods to compute related topological invariants of the subgroup within the group.

Contribution

It presents a novel algorithm for detecting splittings over quasiconvex subgroups in hyperbolic groups, extending previous techniques with new graph-based methods.

Findings

Algorithm successfully decides subgroup splittings.

Provides methods to compute filtered ends and ends of the subgroup.

Extends existing techniques using labelled digraphs and boundary analysis.

Abstract

We present an algorithm which decides whether a given quasiconvex residually finite subgroup $H$ of a hyperbolic group $G$ is associated with a splitting. The methods developed also provide algorithms for computing the number of filtered ends $\tilde e(G,H)$ of $H$ in $G$ under certain hypotheses, and give a new straightforward algorithm for computing the number of ends $e(G,H)$ of the Schreier graph of $H$. Our techniques extend those of Barrett via the use of labelled digraphs, the languages of which encode information on the connectivity of $\partial G - \Lambda H$.