Twisted Conjugacy Classes in Lattices in Semisimple Lie Groups

Abstract

Abstract. Given a group automorphism φ : Γ −→ Γ, one has an action of Γ on itself by φ-twisted conjugacy, namely, g.x = gxφ(g −1 ). The orbits of this action are called φ-conjugacy classes. One says that Γ has the R ∞ -property if there are infinitely many φ-conjugacy classes for every automorphism φ of Γ. In this paper we show that any irreducible lattice in a connected non-compact semi simple Lie group having finite centre and rank at least 2 has the R ∞ -property.

Twisted conjugacy and quasi-isometric rigidity of irreducible lattices in semisimple lie groups

“…(i) γ ki = 1, 1 < i ≤ n, for all k ∈ N, and, (ii) the γ k,1 ∈ Γ 1 are in pairwise distinct ψ-twisted conjugacy classes where ψ := φ r | Γ 1 = φ 1 • φ r • · · · • φ 2 ∈ Aut(Γ 1 ). Since Γ 1 is an irreducible lattice, such a sequence exists by the main theorem of [14] when the real rank of H 1 is at least 2, by [11] when Γ 1 is uniform and the real rank of H 1 equals 1, and by the result of [6] when Γ 1 is nonuniform and the rank of H 1 equals 1.…”

“…We need to show that R(φ) = ∞. By the main result of [14], we may (and do) assume that Λ is reducible.…”

The R∞ property for nilpotent quotients of surface groups

“…We refer to the paper [12] for an overview of the families of groups which have been studied in this context until 2008. After that people have also studied almostcrystallographic groups [2,5], Baumslag-Solitar groups [9], braid groups [8], some classes of metabelian groups [10] and several classes of linear groups [26][27][28][29]. We would also like to point the reader to [14] where several aspects of the R ∞ property are discussed.…”

Twisted Burnside–Frobenius Theory for Endomorphisms of Polycyclic Groups

“…A group is called R ∞ if any its automorphism has infinite Reidemeister number. This was the subject of an intensive recent research and for many groups this property was established, see the following partial bibliography and the literature therein: [7,33,15,16,10,2,25,17,36,34,18,26,40,4,20,28]. In some situations the property R ∞ has some direct topological consequences (see e.g.…”