A twisted Burnside theorem for countable groups and Reidemeister numbers
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, φ one of its automorphisms, R(φ) the number of φ-conjugacy classes, and S(φ) = #Fix(\( \hat \phi \)) the number of φ-invariant equivalence classes of irreducible unitary representations. If one of R(φ) and S(φ) is finite, then it is equal to the other.
This conjecture plays a important role in the theory of twisted conjugacy classes (see , ) and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Noncommutative 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.
KeywordsIrreducible Representation Conjugacy Class Countable Group Point Class Twisted Action
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