Finite non-parabolic subgroups of relatively hyperbolic groups

TL;DR

The paper establishes a concrete, computable bound on the orders of finite non-parabolic subgroups in finitely generated relatively hyperbolic groups given a finite relative presentation. It develops a framework based on relative Dehn functions, isolated components, and norm-decreasing conjugations to show that large non-parabolic finite subgroups must concentrate into peripheral subgroups, yielding an explicit bound |H| ≤ (K^{(ℓ+1)^2})! with ℓ ≤ 4δ+1 and K=(2|X∪Ω|)^{2MC+1}. The bounds depend on C (relative isoperimetric constant), δ (hyperbolicity constant), M (max relator length), and Ω (H-letters in relators), and are computable under the decidability assumptions for peripheral word problems. A corollary provides a universal algorithm for solving the order problem for non-parabolic elements in the class of finitely generated relatively hyperbolic groups under these hypotheses, with a refinement when all peripheral subgroups are torsion-free. Overall, the work extends known hyperbolic-group bounds to the relatively hyperbolic setting and gives constructive tools for effective subgroup analysis and decision problems.

Abstract

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_λ\}_{λ\in Λ}$. 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 $H_λ$, $λ\in Λ$, is decidable.

Theorems & Definitions (21)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Theorem 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• Theorem 2.6
• Corollary 2.7