Advertisement

Proceedings - Mathematical Sciences

, Volume 120, Issue 2, pp 217–241 | Cite as

Splittings of free groups, normal forms and partitions of ends

  • Siddhartha GadgilEmail author
  • Suhas Pandit
Article
  • 48 Downloads

Abstract

Splittings of a free group correspond to embedded spheres in the 3-manifold M = # k S 2 × S 1. These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in π 2(M) can be represented by an embedded sphere.

Keywords

Free groups sphere complex algebraic intersection numbers graphs of trees 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bestvina Mladen, Feighn Mark and Handel Michael, The Tits alternative for Out(F n) I, Dynamics of exponentially-growing automorphisms, Ann. Math. (2) 151(2) (2000) 517–623zbMATHCrossRefGoogle Scholar
  2. [2]
    Bestvina Mladen and Feighn Mark, The topology at infinity of Out(F n), Invent. Math. 140(3) (2000) 651–692zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Bestvina M, Feighn M and Handel M, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7(2) (1997) 215–244zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Bestvina Mladen and Handel Michael, Train tracks and automorphisms of free groups, Ann. Math. (2) 135(1) (1992) 1–51CrossRefGoogle Scholar
  5. [5]
    Bowditch Brian H, Intersection numbers and the hyperbolicity of the complex of curves, J. Reine Angew. Math. 598 (2006) 105129Google Scholar
  6. [6]
    Culler M and Morgan J M, Group actions on R-trees, Proc. London Math. Soc. 55 (1987) 571–604zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Culler M and Vogtmann K, Modulii of graphs and automorphisms of free group, Invent. Math. 87(1) (1986) 91–119CrossRefMathSciNetGoogle Scholar
  8. [8]
    Dunwoody MJ, Accessibility and groups of cohomological dimension one, Proc. London Math. Soc. (3) 38(2) (1979) 193–215zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Gadgil S, Embedded spheres in #n S 2 × S 1, Topology and its Applications 153 (2006) 1141–1151zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Gadgil S and Pandit S, Algebraic and geometric intersection numbers for free groups, Topology and its Applications 156(9) (2009) 1615–1619CrossRefMathSciNetGoogle Scholar
  11. [11]
    Gadgil S and Pandit S, Geosphere Laminations for free groups, preprintGoogle Scholar
  12. [12]
    Harer John L, Stability of the homology of the mapping class groups of orientable surfaces, Ann. Math. (2) 121 (1985) 215–249CrossRefGoogle Scholar
  13. [13]
    Harer John L, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157–176zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Hatcher Allen, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995) 39–62zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Hatcher Allen, Algebraic topology (Cambridge University Press) (2001)Google Scholar
  16. [16]
    Hatcher Allen, On triangulations of surfaces, Topology Appl. 40 (1991) 189–194zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Hatcher Allen and Vogtmann Karen, Isoperimetric inequalities for automorphism groups of free groups, Pacific J. Math. 173 (1996) 425–441zbMATHMathSciNetGoogle Scholar
  18. [18]
    Hatcher Allen and Vogtmann Karen, The complex of free factors of a free group, Quart. J. Math. Oxford Ser. (2) 49 (1998) 459–468zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Hatcher Allen and Vogtmann Karen, Rational homology of Aut (F n), Math. Res. Lett. 5 (1998) 759–780zbMATHMathSciNetGoogle Scholar
  20. [20]
    Hatcher Allen and Vogtmann Karen, Homology stability for outer automorphism groups of free groups, Algebr. Geom. Topol. 4 (2004) 1253–1272zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Hatcher Allan, Vogtmann Karen and Wahl Natalie, Erratum to: Homology stability for outer automorphism groups of free groups, Algebr. Geom. Topol. 4 (2004) 1253–1272, by Hatcher and Vogtmann, Algebr. Geom. Topol. 6 (2006) 573–579CrossRefMathSciNetGoogle Scholar
  22. [22]
    Hatcher Allen and Wahl Nathalie, Stabilization for the automorphisms of free groups with boundaries, Geom. Topol. 9 (2005) 1295–1336zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Hatcher Allen and Wahl Nathalie Erratum to: Stabilization for the automorphisms of free groups with boundaries, Geom. Topol. 9 (2005) 1295–1336, Geom. Topol. 12(2) (2008) 639–641zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Kim Y D, The Thurston boundary of Teichmüller space and curve complex, Topology Appl. 154(3) (2007) 675–682zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Laudenbach Francois, Topologie de la dimension trois: homotopie et isotopie (French) Astérisque, No. 12 (Paris: Société Mathématique de France) (1974)Google Scholar
  26. [26]
    Luo Feng, Automorphisms of the complex of curves, Topology 39 (2000) 283–298zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Lyndon R C and Schupp P E, Combinatorial Group Theory (Berlin Heidelberg New York: Springer-Verlag) (1977)zbMATHGoogle Scholar
  28. [28]
    Luo Feng, A presentaion of the mapping class groups, arXiv.math.GT/9801025, v1, 7 Jan (1998)Google Scholar
  29. [29]
    Luo Feng and Stong R Measured lamination spaces on surfaces and geometric intersection numbers, Topology Appl. 136(1–3) (2004) 205–217zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    Masur H A and Minsky Yair, Geometry of the complex of curves I, Hyperbolicity, Invent. Math. 138 (1999) 103–149zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    Scott Peter and Swarup Gadde A, Splittings of groups and intersection numbers, Geom. Topol. 4 (2000) 179–218zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Department of MathematicsIndian Institute of Science Education and ResearchPuneIndia

Personalised recommendations