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# Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

Published:2012-05-17
Printed: Mar 2014
• T. Mubeena,
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
• P. Sankaran,
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
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## Abstract

Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that $\operatorname{SL}(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $\Gamma$ has the $R_\infty$-property where $\Gamma$ is a torsion free non-elementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}), \operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.
 Keywords: twisted conjugacy classes, hyperbolic groups, lattices in Lie groups
 MSC Classifications: 20E45 - Conjugacy classes