Advertisement

Geometriae Dedicata

, Volume 146, Issue 1, pp 211–223 | Cite as

Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

  • Alexander Fel’shtynEmail author
  • Daciberg L. Gonçalves
Original Paper

Abstract

We prove that the symplectic group \({Sp(2n,\mathbb{Z})}\) and the mapping class group Mod S of a compact surface S satisfy the R property. We also show that B n (S), the full braid group on n-strings of a surface S, satisfies the R property in the cases where S is either the compact disk D, or the sphere S 2. This means that for any automorphism \({\phi}\) of G, where G is one of the above groups, the number of twisted \({\phi}\)-conjugacy classes is infinite.

Keywords

Reidemeister number Twisted conjugacy classes Braids group Mapping class group Symplectic group 

Mathematics Subject Classification (2000)

20E45 37C25 55M20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arthur, J., Clozel, L.: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. Princeton University Press, Princeton (1989) (MR90m:22041)Google Scholar
  2. 2.
    Bleak, C., Fel’shtyn, A., Gonçalves, D.L.: Twisted conjugacy classes. In: R. Thompson’s group F (eds.) Pacific Journal of Mathematics 238, 1–6 (2008)Google Scholar
  3. 3.
    Bowditch B.: Intersection numbers and the hyperbolicity of the curve complex. J. Reine Angew. Math. 598, 105–131 (2006)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bridson M., Vogtmann K.: Automorphisms of automorphism groups of free group. J. Algebra 229, 785–792 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Druţu, C.: Relatively hyperbolic groups: geometry and quasi-isometric invariance. E-print arXiv: math.GR/0605211Google Scholar
  6. 6.
    Dyer J.L., Grossman E.K.: The automorphism groups of the braid groups. Amer. J. Math. 103(6), 1151–1169 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dyer J., Formanek E.: The automorphism group of free group is complete. J. Lond. Math. Soc. II. Ser. 11, 181–190 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fel’shtyn, A.: Dynamical zeta functions, Nielsen theory and Reidemeister torsion. Mem. Amer. Math. Soc. 147(699), (2000) (xii+146. MR2001a:37031)Google Scholar
  9. 9.
    Fel’shtyn A.L.: The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite. Zapiski Nauchnych Seminarov POMI 279, 229–241 (2001)Google Scholar
  10. 10.
    Fel’shtyn, A.L., Gonçalves, D.L.: Reidemeister numbers of any automorphism of Baumslag–Solitar group is infinite. Geometry and Dynamics of Groups and Spaces. Prog Math 265, 286–306 (2008) (Birkhauser)Google Scholar
  11. 11.
    Fel’shtyn A.L., Hill R.: The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion. K-theory 8(4), 367–393 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fel’shtyn A., Troitsky E.: Geometry of Reidemeister classes and twisted Burnside theorem. J K-Theory 2(3), 405–445 (2008)MathSciNetGoogle Scholar
  13. 13.
    Fel’shtyn A., Troitsky E.: Twisted Burnside–Frobenius theory for discrete groups. J. Reine Angew. Math. 613, 193–210 (2007)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Fel’shtyn A.L., Troitsky E.: Twisted Burnside Theorem for Countable Groups and Reidemeister Numbers, Noncommutative Geometry and Number Theory, pp. 141–154. Vieweg, Braunschweig (2006)Google Scholar
  15. 15.
    Fel’shtyn A., Troitsky E., Vershik A.: Twisted Burnside theorem for type II1 groups: an example. Math Res Lett 13(5), 719–728 (2006)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Fel’shtyn A., Leonov Y., Troitsky E.: Reidemeister numbers of saturated weakly branch groups. Geometria Dedicata 134, 61–73 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fel’shtyn, A.: New directions in Nielsen–Reidemeister theory. (E-print arxiv: math.GR/0712.2601)Google Scholar
  18. 18.
    Gonçalves D.L.: The coincidence Reidemeister classes on nilmanifolds and nilpotent fibrations. Top. Appl. 83, 169–186 (1998)zbMATHCrossRefGoogle Scholar
  19. 19.
    Gonçalves, D.L., Guaschi, J.: The lower central and derived series of the braid groups B n(S 2) and \({B_m(S^2\setminus\{x_1,\ldots,x_n\})}\), preprint, March (2006)Google Scholar
  20. 20.
    Gonçalves D.L., Wong P.: Twisted conjugacy classes in wreath products. Int. J. Algebra Comput. 16(5), 875–886 (2006)zbMATHCrossRefGoogle Scholar
  21. 21.
    Grothendieck, A.: Formules de Nielsen–Wecken et de Lefschetz en géométrie algébrique, Séminaire de Géométrie Algébrique du Bois-Marie 1965–1966. SGA 5. Lecture Notes in Mathematics, vol. 569, Springer, Berlin, pp. 407–441 (1977)Google Scholar
  22. 22.
    Hua L.K., Reiner I.: Automorphisms of the unimodular group. Trans. Amer. Math. Soc. 71, 331–348 (1951)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Ivanov N.V.: Automorphisms of Teichm uller Modular Groups, Lecture Notes in Mathematics 1346. Springer, Berlin (1988)Google Scholar
  24. 24.
    Jiang B.: Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, vol 14. American Mathematical Society, Providence (1983)Google Scholar
  25. 25.
    Khramtsov, D.G.: Completeness of groups of outer automorphisms of free groups. In Group-Theoretic Investigations(Russian), pp. 128–143, Akad. Nauk SSSR Ural. Otdel., Sverdlovsk. (1990)Google Scholar
  26. 26.
    Levitt, G.: On the automorphism group of generalised Baumslag–Solitar groups. (2005) ((E-print arxiv:math.GR/0511083))Google Scholar
  27. 27.
    Levitt G., Lustig M.: Most automorphisms of a hyperbolic group have very simple dynamics. Ann. Scient. Éc. Norm. Sup. 33, 507–517 (2000)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Magnus W., Karrass A., Solitar D.: Combinatorial Group Theory. Interscience Publishers, New York (1966)zbMATHGoogle Scholar
  29. 29.
    Masur H.A., Minsky Y.N.: Geometry of the complex of curves I: hyperbolicity. Invent. Math. 138, 103–149 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Newmann M.: Integral Matrices. Academic Press, New York (1972)Google Scholar
  31. 31.
    Reidemeister K.: Automorphismen von Homotopiekettenringen. Math. Ann. 112, 586–593 (1936)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Reiner I.: Automorphisms of the symplectic modular group. Trans. Amer. Math. Soc. 80, 35–50 (1955)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Scott G.P.: Braid groups and the group of homeomorphisms of a surface. Proc. Camb. Phil. Soc. 68, 605–617 (1970)zbMATHCrossRefGoogle Scholar
  34. 34.
    Shokranian, S.: The Selberg–Arthur trace formula, vol 1503. Lecture Notes in Mathematics. Springer, Berlin. Based on lectures by James Arthur. MR1176101 (93j:11029) (1992)Google Scholar
  35. 35.
    Taback, J., Wong, P.: Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag–Solitar groups (2006) (E-print arxiv:math.GR/0601271)Google Scholar
  36. 36.
    Taback, J., Wong, P.: A note on twisted conjugacy and generalized Baumslag–Solitar groups. (2006) (E-print arXiv:math.GR/0606284)Google Scholar
  37. 37.
    Troitsky, E.: Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy. Funct. Anal. Pril. 40(2), 44–54 (2006) [In Russian, English translation: Funct. Anal. Appl. 40(2), 117–125 (2006) (Preprint 86 (2004), Max-Planck-Institut für Mathematik, math.OA/0606191)]Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Alexander Fel’shtyn
    • 1
    • 2
    Email author
  • Daciberg L. Gonçalves
    • 3
  1. 1.Instytut MatematykiUniwersytet SzczecinskiSzczecinPoland
  2. 2.Boise State UniversityBoiseUSA
  3. 3.Department de MatemáticaIME, USPSão PauloBrazil

Personalised recommendations