Twisted conjugacy classes in abelian extensions of certain linear groups

Abstract

Given a group automorphism ø:Γ→Γ, one has an action of Γ on itself by ø-twisted conjugacy, namely,g.x=gxøg^-1). The orbits of this action are called ø-twisted conjugacy classes. One says that has the R∞-property if there are infinitely many ø-twisted conjugacy classes for every automorphism ø of Γ. In this paper we show that SL(n,Z) and its congruence subgroups have the R∞-property. Further we show that any (countable) abelian extension of Γ has the R∞-property where Γ is a torsion free non-elementary hyperbolic group, or SL(n,Z),Sp,(2n,Z) or a principal congruence subgroup of SL(n,Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.