Finite non-parabolic subgroups of relatively hyperbolic groups

TL;DR

This paper establishes an upper bound on the orders of finite non-parabolic subgroups in relatively hyperbolic groups, which is computable under certain conditions, advancing understanding of subgroup structures in geometric group theory.

Contribution

It provides a computable upper bound on finite non-parabolic subgroup orders in relatively hyperbolic groups based on their finite relative presentation.

Findings

Upper bound on subgroup orders expressed in terms of presentation constants

Bound is computable if the group is finitely generated and the word problem is decidable in peripheral subgroups

Advances understanding of subgroup structure in relatively hyperbolic groups

Abstract

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_{\lambda}\}_{\lambda\in \Lambda}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We give an upper bound on the orders of finite non-parabolic subgroups of $G$ in terms of some fundamental constants associated with $\mathcal{P}$. This upper bound is computable if $G$ is finitely generated and the word problem in each $H_{\lambda}$, $\lambda\in \Lambda$, is decidable.