Geometriae Dedicata

, Volume 92, Issue 1, pp 95–103 | Cite as

The Subgroups of Direct Products of Surface Groups

  • Martin R. Bridson
  • James Howie
  • Charles F. MillerIII
  • Hamish Short


A subgroup of a product of n surface groups is of type FPn if and only if it contains a subgroup of finite index that is itself a product of (at most n) surface groups.

subgroup direct product free groups surface groups homology of groups 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bieri, R.: Normal subgroups in duality groups and in groups of cohomological dimension 2, J. Pure Appl. Algebra 7 (1976), 35–51.Google Scholar
  2. 2.
    Bridson, M. R. and Wise, D. T.: VH complexes, towers and subgroups of F x F, Math. Proc. Cambridge Philos. Soc. 126 (1999), 481–497.Google Scholar
  3. 3.
    Brown, K. S.: Cohomology of Groups, Grad. Texts in Math. 87, Springer-Verlag, Berlin, 1982.Google Scholar
  4. 4.
    Baumslag, G. and Roseblade, J. E.: Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1984), 44–52.Google Scholar
  5. 5.
    Carr, N. C.: Complex flat manifolds and their moduli spaces, PhD thesis, University College London (1982).Google Scholar
  6. 6.
    Grunewald, F.: On some groups which cannot be finitely presented, J. London Math. Soc. (2) 17 (1978), 427–436.Google Scholar
  7. 7.
    Hall Jr., M.: Subgroups of finite index in free groups, Canad. J. Math. 1 (1949), 187–190.Google Scholar
  8. 8.
    Johnson, F. E. A.: Subgroups of products of surface groups, Math. Proc. Cambridge Philos. Soc. 126 (1999), 195–208.Google Scholar
  9. 9.
    Johnson, F. E. A.: Finitely presented normal subgroups of a product of Fuchsian groups, J. Algebra 231(1) (2000), 39–52.Google Scholar
  10. 10.
    Massey, W. S.: Algebraic Topology: An Introduction, Grad. Texts in Math. 56, Springer-Verlag, Berlin, 1977.Google Scholar
  11. 11.
    Meinert, H.: The geometric invariants of direct products of virtually free groups, Comment. Math. Helv. 69 (1994), 39–48.Google Scholar
  12. 12.
    Scott, G. P.: Subgroups of surface groups are almost geometric, J. London Math Soc. (2) 17 (1978), 555–565 and correction in J. London Math Soc (2) 32 (1985), 217–220.Google Scholar
  13. 13.
    Short, H.: Finitely presented subgroups of a product of free groups, Oxford Quart. J. 52 (2001), 127–131.Google Scholar
  14. 14.
    Stallings, J. R.: A finitely presented group whose 3-dimensional homology group is not finitely generated, Amer. J. Math. 85 (1963), 541–543.Google Scholar
  15. 15.
    Stallings, J. R.: The topology of finite graphs, Invent. Math. 71 (1983), 551–565.Google Scholar
  16. 16.
    Stillwell, J.: Classical Topology and Combinatorial Group Theory, Grad. Texts in Math. 72, Springer-Verlag, Berlin, 1980.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Martin R. Bridson
    • 1
  • James Howie
    • 2
  • Charles F. MillerIII
    • 3
  • Hamish Short
    • 4
  1. 1.Mathematical InstituteOxfordU.K.
  2. 2.Department of MathematicsHeriot-Watt UniversityEdinburghU.K.
  3. 3.Department of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  4. 4.LATP, UMR 6632Université de ProvenceMarseille cedex 13France

Personalised recommendations