Geometriae Dedicata

, Volume 114, Issue 1, pp 49–70

Automorphisms of Hyperbolic Groups and Graphs of Groups



Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups Ge are rigid (i.e. Out(Ge) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.


automorphism groups graphs of groups hyberbolic groups mapping class groups JSJ decomposition tree automorphisms 

Mathematics Subject Classifications (2000)

20F65 20E08 20F67 20F28 20E06 


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  1. 1.
    Bass, H., Jiang, R. 1996Automorphism groups of tree actions and of graphs of groupsJ. Pure Appl. Algebra112109155CrossRefGoogle Scholar
  2. 2.
    Bestvina, M., Feighn, M. 1995Stable actions of groups on real treesInvent. Math.121287321CrossRefGoogle Scholar
  3. 3.
    Bestvina, M., Handel, M. 1992Train tracks for automorphisms of the free groupAnn. Math.135151Google Scholar
  4. 4.
    Bowditch, B. 1998Cut points and canonical splittings of hyperbolic groupsActa Math.180145186Google Scholar
  5. 5.
    Collins, D. J., Turner, E. C. 1994Efficient representatives for automorphisms of free productsMichigan Math. J.41443464CrossRefGoogle Scholar
  6. 6.
    Dicks, W. and Formanek, E.: Algebraic mapping-class groups of orientable surfaces with boundaries, Preprint (2003).Google Scholar
  7. 7.
    Fujiwara, K. 2002On the outer automorphism group of a hyperbolic groupIsrael J. Math.131277284Google Scholar
  8. 8.
    Guirardel, V. and Levitt, G.: In preparation.Google Scholar
  9. 9.
    Harvey, W. J., Maclachlan, C. 1975On mapping-class groups and Teichmüller spacesProc. London Math. Soc.30496512Google Scholar
  10. 10.
    Ivanov, N. V., McCarthy, J. D. 1999On injective homomorphisms between Teichmüller modular groups IInvent. Math.135425486CrossRefGoogle Scholar
  11. 11.
    Jensen, C. A., Wahl, N. 2004Automorphisms of free groups with boundariesAlgebr. Geom. Topol.4543569CrossRefGoogle Scholar
  12. 12.
    Levitt, G.: The outer space of a free product, Scholar
  13. 13.
    Levitt, G. and Lustig, M.: Automorphisms of free groups have asymptotically periodic dynamics, Preprint.Google Scholar
  14. 14.
    Miller III, C. F., Neumann, W. D. and Swarup, G. A.: Some examples of hyperbolic groups, In: Geometric Group Theory Down Under, de Gruyter, Berlin, 1999, pp. 195–202.Google Scholar
  15. 15.
    Paulin, F. 1991

    Outer automorphisms of hyperbolic groups and small actions on R-trees

    Alperin, R. C. eds. Arboreal Group TheorySpringerNew York331343MSRI Publ. 19
    Google Scholar
  16. 16.
    Pettet, M. R. 1997Virtually free groups with finitely many outer automorphismsTrans. Amer. Math. Soc.34945654587CrossRefGoogle Scholar
  17. 17.
    Rips, E., Sela, Z. 1994Structure and rigidity in hyperbolic groups IGeom. Funct. Anal.4337371CrossRefGoogle Scholar
  18. 18.
    Sela, Z. 1997Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups IIGeom. Funct. Anal.7561593CrossRefGoogle Scholar
  19. 19.
    Shor J.: A Scott conjecture for hyperbolic groups, Preprint (1999).Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.LMNO, UMR CNRS 6139Université de CaenCaen CedexFrance

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