Twisted conjugacy classes in residually finite groups

Abstract

We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional irreducible unitary representations being invariant for the dual of this automorphism. Also, we prove that any finitely generated residually finite non-amenable group has the R-infinity property (any automorphism has infinitely many twisted conjugacy classes). This gives a lot of new examples and covers many known classes of such groups.

20 pages, no figures, v2: typos corrected, references added

References in corpus (6)

• Rank gradient and torsion groups
• Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
• Kazhdan quotients of Golod-Shafarevich groups
• Holonomy groups of flat manifolds with $R_{\infty }$ property
• A twisted Burnside theorem for countable groups and Reidemeister numbers
• Twisted conjugacy classes in special and general linear groups

Cited by in corpus (3)

• Twisted conjugacy in soluble arithmetic groups
• The $R_{\infty }$-property for Chevalley groups of types $B_l,C_l,D_l$ over integral domains
• Twisted conjugacy classes of the unit element