Twisted Conjugacy for Virtually Cyclic Groups and Crystallographic Groups

  • Daciberg Gonçalves
  • Peter Wong
Part of the Trends in Mathematics book series (TM)


A group is said to have the property R if every automorphism has an infinite number of twisted conjugacy classes. In this paper, we classify all virtually cyclic groups with the R property. Furthermore, we determine which of the 17 crystallographic groups of rank 2 have this property.

Mathematics Subject Classification (2000)

Primary: 20E45 Secondary: 55M20 


Reidemeister number elementary groups Gromov hyperbolic groups fixed point theory 


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  1. [1]
    H.S.M. Coxeter and W.O.J. Moser, Generators and relations for discrete groups. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957. viii + 155 pp.zbMATHGoogle Scholar
  2. [2]
    W. Dicks and M. Dunwoody, Groups acting on graphs. Cambridge Studies in Advanced Mathematics, 17. Cambridge University Press, Cambridge, 1989. xvi+283 pp.Google Scholar
  3. [3]
    A.L. Fel’shtyn, The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite, Zapiski Nauchnych Seminarov POMI 279 (2001), 229–241.Google Scholar
  4. [4]
    D. Gonçalves and P. Wong, Twisted conjugacy classes in exponential growth groups, Bull. London Math. Soc. 35 (2003), 261–268.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    D. Gonçalves and P. Wong, Twisted conjugacy classes for nilpotent groups, J. Reine Angew. Math. 633 (2009), 11–27.zbMATHMathSciNetGoogle Scholar
  6. [6]
    B. Jiang, “Lectures on Nielsen Fixed Point Theory,” Contemporary Mathematics vol. 14, Amer. Math. Soc., Providence, Rhode Island, 1983.Google Scholar
  7. [7]
    G. Levitt and M. Lustig, Most automorphisms of a hyperbolic group have very simple dynamics, Ann. Scient. Éc. Norm. Sup. 33 (2000), 507–517.zbMATHMathSciNetGoogle Scholar
  8. [8]
    R. Lyndon, Groups and Geometry. LMN Lecture Note Series, 101, Cambridge University press, 1987.Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Daciberg Gonçalves
    • 1
  • Peter Wong
    • 2
  1. 1.Dept. de Matemática — IME — USPSão PauloBrasil
  2. 2.Department of MathematicsBates CollegeLewistonUSA

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