Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

Abstract

Given an automorphism $\varphi :\Gamma \to \Gamma$, one has an action of $\Gamma$ on itself by $\varphi$-twisted conjugacy, namely, $g.x=gx\varphi (g^{-1})$. The orbits of this action are called $\varphi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_{\infty }$-property if there are infinitely many $\varphi$-twisted conjugacy classes for every automorphism $\varphi$ of $\Gamma$. In this paper we show that 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 SL$(n,\mathbb{Z})$, Sp$(2n,\mathbb{Z})$ or a principal congruence subgroup of SL$(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Cited by in corpus (11)

• Twisted Conjugacy Classes in Lattices in Semisimple Lie Groups
• Twisted conjugacy classes in residually finite groups
• Structure and automorphisms of pure virtual twin groups
• Twisted Conjugacy in Direct Products of Groups
• The $R_{\infty }$-property for right-angled Artin groups
• Some remarks on twin groups
• Twisted conjugacy classes in twisted Chevalley groups
• Automorphisms of odd Coxeter groups
• Reidemeister classes in some weakly branch groups
• The $R_{\infty }$ and $S_{\infty }$ properties for linear algebraic groups
• Twisted conjugacy in classical groups over certain domains of Characteristic $p>0$