Geometriae Dedicata

, Volume 147, Issue 1, pp 29–45 | Cite as

Profinite properties of graph manifolds

Original Paper

Abstract

Let M be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π1(M) is efficient with respect to the JSJ decomposition of M. We go on to prove that π1(M) is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if M is a graph manifold then π1(M) is conjugacy separable.

Keywords

3-Manifolds Profinite groups Conjugacy separability 

Mathematics Subject Classification (2000)

20E26 57N10 

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References

  1. 1.
    Agol I., Long D.D., Reid A.W.: The Bianchi groups are separable on geometrically finite subgroups. Ann. Math. Second Series 153, 599–621 (2001)MATHMathSciNetGoogle Scholar
  2. 2.
    Burns R.G., Karrass A., Solitar D.: A note on groups with separable finitely generated subgroups. Bull. Aust. Math. Soc. 36, 153–160 (1987)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chagas S.C., Zalesskii P.A.: The figure eight knot group is conjugacy separable. J. Algebra Appl. 8, 539–556 (2009)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chagas S.C., Zalesskii P.A.: Finite index subgroups of conjugacy separable groups. Forum Math. 21, 347–353 (2009)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dicks W.: Groups, Trees and Projective Modules, Volume 790 of Lecture Notes in Mathematics. Springer, Berlin (1980)Google Scholar
  6. 6.
    Goryaga A.V.: Example of a finite extension of an FAC-group that is not an FAC-group. Akademiya Nauk SSSR. Sibirskoe Otdelenie. Sib. Mat. Z. 27, 203–205, 225 (1986)MATHMathSciNetGoogle Scholar
  7. 7.
    Grunewald F., Jaikin-Zapirain A., Zalesskii P.A.: Cohomological properties of the profinite completion of Kleinian groups. Duke Math. J. 144, 53–72 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hamilton E.: Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic n-orbifolds. Proc. Lond. Math. Soc. Third Series 83, 626–646 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hempel J.: Residual Finiteness for 3-manifolds, Volume 111 of Annals of Mathematical Studies, pp. 379–396. Princeton University Press, Princeton (1987)Google Scholar
  10. 10.
    Jaco W.H., Shalen P.B.: Seifert fibered spaces in 3-manifolds. Mem. Am. Math. Soc. 21, viii + 192 (1979)MathSciNetGoogle Scholar
  11. 11.
    Johannson K.: Homotopy Equivalences of 3-manifolds with Boundaries, Volume 761 of Lecture Notes in Mathematics. Springer, Berlin (1979)Google Scholar
  12. 12.
    Long D.D., Niblo G.A.: Subgroup separability and 3-manifold groups. Math. Z. 207, 209–215 (1991)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Martino, A.: A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable. J. Algebra 313, 773–781, (2007). arXiv:math/0505565, May 2005Google Scholar
  14. 14.
    Neumann, W.D., Swarup, G.A.: Canonical decompositions of 3-manifolds. Geom. Topol. 1, 21–40 (electronic) (1997)Google Scholar
  15. 15.
    Niblo G.A.: Separability properties of free groups and surface groups. J. Pure Appl. Algebra 78, 77–84 (1992)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Niblo G.A., Wise D.T.: Subgroup separability, knot groups and graph manifolds. Proc. Am. Math. Soc. 129, 685–693 (2001)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. ArXiv:math/0211159, November 2002Google Scholar
  18. 18.
    Perelman, G.: Finite extinction time for the solutions to the ricci flow on certain three-manifolds. ArXiv:math/0307245, July 2003Google Scholar
  19. 19.
    Perelman, G.: Ricci flow with surgery on three-manifolds. ArXiv:math/0303109, March 2003Google Scholar
  20. 20.
    Préaux J.-P.: Conjugacy problem in groups of oriented geometrizable 3-manifolds. Topology 45, 171–208 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Remeslennikov V.N.: Conjugacy in polycyclic groups. Akademiya Nauk SSSR. Sibirskoe Otdelenie. Institut Matematiki. Algebra i Logika 8, 712–725 (1969)MathSciNetGoogle Scholar
  22. 22.
    Ribes L., Zalesskii P.A.: Conjugacy separability of amalgamated free products of groups. J. Algebra 179, 751–774 (1996)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Ribes L., Segal D., Zalesskii P.A.: Conjugacy separability and free products of groups with cyclic amalgamation. J. Lond. Math. Soc. Second Series 57, 609–628 (1998)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Scott P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)MATHCrossRefGoogle Scholar
  25. 25.
    Serre, J.-P.: Arbres, amalgames, SL 2. Société Mathématique de France, Paris (1977). Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46Google Scholar
  26. 26.
    Serre, J.-P.: Galois Cohomology. Springer-Verlag, Berlin (1997) Translated from the French by Patrick Ion and revised by the authorGoogle Scholar
  27. 27.
    Stebe P.F.: A residual property of certain groups. Proc. Am. Math. Soc. 26, 37–42 (1970)MATHMathSciNetGoogle Scholar
  28. 28.
    Stebe P.F.: Conjugacy separability of the groups of hose knots. Trans. Am. Math. Soc. 159, 79–90 (1971)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Thurston W.P.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Am. Math. Soc. Bull. New Series 6, 357–381 (1982)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Zalesskii P.A.: Geometric characterization of free constructions of profinite groups. Sib. Math. J. 30, 73–84, 226 (1989)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Zalesskii P.A., Mel’nikov O.V.: Subgroups of profinite groups acting on trees. Math. USSR Sbornik 63, 405–424 (1989)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Zalesskii, P.A., Mel’nikov, O.V.: Fundamental groups of graphs of profinite groups. Algebra i Analiz 1, 117–135 (1989). Translated in Leningrad Math. J. 1, 921–940 (1990)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mathematics 253-37CaltechPasadenaUSA
  2. 2.Department of MathematicsUniversity of BrasiliaBrasiliaBrazil

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