Twisted Conjugacy classes in lattices in semisimple lie groups [HBNI Th63]

Abstract

Let G be a group and let Ø : Γ → Γ be an endomorphism. We define an action g.x := gxØ(g-1), for g,x ε Γ, of Γ on itself. The Ø-twisted conjugacy class of an element x ε Γ is the orbit of this action containing x. A group Γ has the R∞ -property if every automorphism Ø of Γ has infinitely many Ø-twisted conjugacy classes. In this thesis it is shown that any irreducible lattice in a non-compact connected semisimple Lie group with finite center and having real rank at least 2 has the R∞ -property. It is also shown that any countable abelian extensions Λ of Γ has the R∞-property when (i) the lattice Γ is linear, (ii) Γ is a torsion free non-elementary hyperbolic group. Also considered, the R∞-problem for S -arithmetic lattices.