# From local to global conjugacy of subgroups of relatively hyperbolic
groups

Research paper by **Oleg Bogopolski, Kai-Uwe Bux**

Indexed on: **05 May '16**Published on: **05 May '16**Published in: **Mathematics - Group Theory**

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#### 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 $H_2$ has a loxodromic element. 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 $G$.