Presenting parabolic subgroups

TL;DR

This paper proves that finitely presented relatively hyperbolic groups have finitely presented parabolic subgroups and provides algorithms to find presentations of these subgroups from the larger group's presentation.

Contribution

It establishes the finite presentability of parabolic subgroups in finitely presented relatively hyperbolic groups and introduces algorithms to compute their presentations.

Findings

Finitely presented relatively hyperbolic groups have finitely presented parabolic subgroups.

Algorithms are provided to derive presentations of parabolic subgroups.

An algorithm to find parabolic subgroups in recursively enumerable classes is developed.

Abstract

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its word problem, and generating sets of the parabolic subgroups. We also give an algorithm that finds parabolic subgroups in a given recursively enumerable class of groups.