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Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

Published online by Cambridge University Press:  20 November 2018

T. Mubeena
Affiliation:
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India e-mail: mubeena@imsc.res.in sankaran@imsc.res.in
P. Sankaran
Affiliation:
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India e-mail: mubeena@imsc.res.in sankaran@imsc.res.in
Corresponding
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Abstract

Given a group automorphism $\phi :\,\Gamma \,\to \,\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 $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ 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 $\text{SL(}n\text{,}\mathbb{Z}\text{)},\text{Sp(2}n\text{,}\mathbb{Z}\text{)}$ or a principal congruence subgroup of $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

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