Twisted Burnside theorem for type II${}_1$ groups: an example

Abstract

The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem: Let G be a countable discrete group, f its automorphism, R(f) the number of f-conjugacy classes (Reidemeister number), S(f):=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history and has very serious consequences in Dynamics, while its proof needs rather fine results from Functional and Non-commutative Harmonic Analysis. It was proved for finitely generated groups of type I in a previous paper. In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number.