A twisted Burnside theorem for countable groups and Reidemeister numbers

  • Alexander Fel’shtyn
  • Evgenij Troitsky
Part of the Aspects of Mathematics book series (ASMA)


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 [8], 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 [12], [6]) 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.


Irreducible Representation Conjugacy Class Countable Group Point Class Twisted Action 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

Authors and Affiliations

  • Alexander Fel’shtyn
    • 1
    • 2
  • Evgenij Troitsky
    • 3
  1. 1.Fachbereich Mathematik, Emmy-Noether-CampusUniversität SiegenSiegenGermany
  2. 2.Instytut MatematykiUniwersytet SzczecinskiSzczecinPoland
  3. 3.Dept. of Mech. and Math.Moscow State UniversityMoscowRussia

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