A twisted Burnside theorem for countable groups and Reidemeister numbers

TL;DR

This paper proves a generalized Burnside theorem for finitely generated groups of type I, linking Reidemeister numbers and fixed points of automorphisms, with implications for dynamics and harmonic analysis.

Contribution

It establishes a new relation between Reidemeister numbers and fixed points of automorphisms for a class of countable groups, extending classical results.

Findings

Proved the conjecture for finitely generated type I groups.

Established equality between Reidemeister number and fixed point count when either is finite.

Presented applications and examples illustrating the theorem's implications.

Abstract

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and 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 an important role in the theory of twisted conjugacy classes and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Non-commutative Harmonic Analysis. We begin a discussion of the general case (which needs another definition of the dual object). It will be the subject of a forthcoming paper. Some applications and examples are presented.