From local to global conjugacy of subgroups of relatively hyperbolic groups

TL;DR

This paper investigates the conjugacy relations between subgroups in relatively hyperbolic groups, establishing conditions under which elementwise conjugacy implies conjugacy of finite index subgroups, with applications to limit groups.

Contribution

It extends the understanding of subgroup conjugacy in relatively hyperbolic groups, providing new criteria and estimates for conjugators, especially in the context of limit groups.

Findings

Finite index subgroup of H2 conjugate into H1

Minimal conjugator length can be estimated

Results apply to limit groups with relaxed conditions

Abstract

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\mathbb{P}=\{P_1,\dots,P_m\}$. Let $H_1,H_2$ be subgroups of $G$ such that $H_1$ is relatively quasiconvex with respect to $\mathbb{P}$ and $H_2$ is not parabolic. Suppose that $H_2$ is elementwise conjugate into $H_1$. Then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. The minimal length of the conjugator can be estimated. In the case where $G$ is a limit group, it is sufficient to assume only that $H_1$ is a finitely generated and $H_2$ is an arbitrary subgroup of $G$.