Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

TL;DR

This paper proves that certain linear groups and their abelian extensions have infinitely many twisted conjugacy classes for every automorphism, highlighting a property related to the structure and symmetry of these groups.

Contribution

It establishes the $R_ty$-property for SL(n,Z), its congruence subgroups, and their abelian extensions over various hyperbolic and geometric groups.

Findings

SL(n,Z) has the $R_ty$-property.

Abelian extensions of specified hyperbolic and linear groups have the $R_ty$-property.

The property holds for fundamental groups of negatively curved Riemannian manifolds.

Abstract

Given an 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 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.