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 


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  1. [1]
    James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Princeton University Press, Princeton, NJ, 1989. MR 90m:22041MATHGoogle Scholar
  2. [2]
    Alain Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994. MR 95j:46063MATHGoogle Scholar
  3. [3]
    J. Dixmier, C*-algebras, North-Holland, Amsterdam, 1982.Google Scholar
  4. [4]
    P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. math. France 92 (1964), 181–236.MATHMathSciNetGoogle Scholar
  5. [5]
    A. L. Fel’shtyn, The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), no. Geom. i Topol. 6, 229–240, 250. MR 2002e:20081Google Scholar
  6. [6]
    Alexander Fel’shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion, Mem. Amer. Math. Soc. 147 (2000), no. 699, xii+146. MR 2001a:37031MathSciNetGoogle Scholar
  7. [7]
    Alexander Fel’shtyn and Daciberg Gonçalves, Twisted conjugacy classes of automorphisms of Baumslag-Solitar groups, http://de.arxiv.org/abs/math.GR/0405590.Google Scholar
  8. [8]
    Alexander Fel’shtyn and Richard Hill, The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion, K-Theory 8 (1994), no. 4, 367–393. MR 95h:57025CrossRefMathSciNetGoogle Scholar
  9. [9]
    —, Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion, Nielsen theory and Reidemeister torsion (Warsaw, 1996), Polish Acad. Sci., Warsaw, 1999, pp. 77–116. MR 2001h:37047Google Scholar
  10. [10]
    Daciberg Gonçcalves and Peter Wong, Twisted conjugacy classes in exponential growth groups, Bull. London Math. Soc. 35 (2003), no. 2, 261–268. MR 2003j:20054CrossRefMathSciNetGoogle Scholar
  11. [11]
    A. Grothendieck, Formules de Nielsen-Wecken et de Lefschetz en géométrie algébrique, Séminaire de Géométrie Algébrique du Bois-Marie 1965-66. SGA 5, Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977, pp. 407–441.CrossRefGoogle Scholar
  12. [12]
    B. Jiang, Lectures on Nielsen fixed point theory, Contemp. Math., vol. 14, Amer. Math. Soc., Providence, RI, 1983.Google Scholar
  13. [13]
    A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, Berlin Heidelberg New York, 1976.MATHGoogle Scholar
  14. [14]
    Salahoddin Shokranian, The Selberg-Arthur trace formula, Springer-Verlag, Berlin, 1992, Based on lectures by James Arthur. MR 93j:11029MATHGoogle Scholar
  15. [15]
    Elmar Thoma, Über unitäre Darstellungen abzählbarer, diskreter Gruppen, Math. Ann. 153 (1964), 111–138. MR 28 #3332MATHCrossRefMathSciNetGoogle Scholar

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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