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The Hopf Property for Subgroups of Hyperbolic Groups

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Abstract

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).

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References

  1. Lyndon, R. C. and Shupp, P. E.: Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.

    Google Scholar 

  2. Ceccherini-Silberstein, T. G., Grigorchuk, R. I. and Scarabotti, F.: Analytical methods in group theory: hopfianity of free groups, Preprint.

  3. Mal'cev, B.: On the faithful representation of infinite groups by matrices, Math. Sb. (N.S.) 8(50) (1940), 405–422; English translation in Amer. Math. Soc. Transl. 45(2) (1965),1-18.

    Google Scholar 

  4. Geoghegan, R., Mihalik, M., Sapir M. and Wise, D.: Ascending HNN extensions of finitely generated free groups are Hopfian, Proc. London. Math. Soc. 33 (2001), 292–298.

    Google Scholar 

  5. Neumann, B.: A two-generator group isomorphic to a proper factor group, J. London. Math. Soc. 25 (1950), 247–248.

    Google Scholar 

  6. Higman, G.: A finitely related group with an isomorphic proper factor group, J. London. Math. Soc. 26 (1951), 59–61.

    Google Scholar 

  7. Baumslag, G. and Solitar, D.; Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201.

    Google Scholar 

  8. Kurosh, A. G.: The Theory of Groups, 3rd edn, 1967, (in Russian).

  9. Meier, D.: Non-Hopfian groups, J. London Math. Soc. 26(2) (1982), 265–270.

    Google Scholar 

  10. Wise, D.: A non-Hopfian automatic group, J. Algebra 180 (1996), 845–847.

    Google Scholar 

  11. Sela, Z. Endomorphisms of hyperbolic groups I: The Hopf property, Topology 38(2) (1999), 301–321.

    Google Scholar 

  12. Rips, E.: Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), 45–47.

    Google Scholar 

  13. Sela, Z.: Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, GAFA 7 (1997), 561–593.

  14. Kapovich, I. and Wise, D.: On the failure of the co-Hopf property for subgroups of wordhyperbolic groups), Israel J. Math. 122(2001), 125–147.

    Google Scholar 

  15. Bestvina, M.: Degeneration of hyperbolic space, Duke Math. J. 56 (1988), 143–161.

    Google Scholar 

  16. Paulin, F.: Outer automorphisms of hyperbolic groups and small actions on R-trees, In: R. C. Alperin (ed.), Arboreal Group Theory, MSRI Publ. 19, Amer. Math. Soc., Providence, 1991, pp. 331–343.

    Google Scholar 

  17. Paulin, F.: Topologie de Gromové quivariante, structures hyperboliques et arbres ré els,Invent. Math. 94 (1988), 53–80.

    Google Scholar 

  18. Bestvina, M. and Feighn, M.: Stable actions of groups on real trees, Invent. Math. 121 (1995), 287–321.

    Google Scholar 

  19. Rips, E. and Sela, Z.: Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997), 53–104.

    Google Scholar 

  20. Sela, Z.: Acylindrical accessibility for groups, Invent. Math. 129(1997), 527–565.

    Google Scholar 

  21. Alonso, J. M., Brady, T., Cooper, D. et al.: Notes on word hyperbolic groups, In: E. Ghys, A. Haefliger and A. Verjovsky (eds), Group Theory from a Geometrical Viewpoint, ICTP, Trieste, 1990, pp. 3-63.

  22. Coornaert, M., Delzant, T. and Papadopoulos, A.: Geometrie et theorie des groupes, Lecture Notes in Math. 1441, Springer-Verlag, New York, 1990.

    Google Scholar 

  23. Ol'shanskii, A. Yu.: Hyperbolic groups, manuscript.

  24. Bridson, M. R. and Swarup, G. A.: On Hausdorff-Gromov convergence and a theorem of Paulin, Enseign Math. 40(1994), 267–289.

    Google Scholar 

  25. Gromov, M.: Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53–78.

    Google Scholar 

  26. Paulin, F.: Actions de groupes sur les arbres, Sémi Bourbaki 94 (1997), 97–137.

    Google Scholar 

  27. Gaboriau, D., Levitt, G. and Paulin, F.: Pseudogroups of isometries of R and Rips' theorem on free actions on R-trees, Israel. J. Math. 87 (1994), 403–428.

    Google Scholar 

  28. Guirardel, V. Approximations of stable actions on R-trees, Comment. Math. Helv. 73 (1998), 89–121.

    Google Scholar 

  29. Bestvina, M.: R-trees in topology, geometry, and group theory, Preprint.

  30. Kapovich, M.: Hyperbolic Manifolds and Discrete Groups: Lectures on Thurston's Hyperbolization, Progr. in Math., Birkhäuser, Basel, 2000.

    Google Scholar 

  31. Rips, E.: Cyclic splittings of finitely presented groups and the canonical JSJ-decomposition Proc. ICM Zurich, 1994, pp. 595-600.

  32. Dunwoody, M. J. and Sageev, M. E.: JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999), 25–44.

    Google Scholar 

  33. Fujiwara, K. and Papasoglu, P.: JSJ-decompositions of finitely presented groups and complexes of groups, Preprint.

  34. Bowditch, B. H.: Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1988), 145–186.

    Google Scholar 

  35. Forester, M.: On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv., in press.

  36. Serre, J. P. Trees, Springer-Verlag, Berlin, 1980.

  37. Gromov, M.: Hyperbolic groups, In: S. M. Gersten (ed.), Essays in Group Theory, MSRI Ser. 8, Springer-Verlag, New York, pp. 75-263.

  38. Higman, G., Neumann, B. H. and Neumann, H.: Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254.

    Google Scholar 

  39. Kargapolov, M. and Merzliakov, Yu.: Fundamentals of the Theory of Groups, Springer-Verlag, New York, 1979 (English edition).

    Google Scholar 

  40. Rips, E. and Sela, Z.: Structure and rigidity in hyperbolic groups I, GAFA 4 (1994), 337–371.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805, Sherbrooke Street West, Montreal, Que., Canada H3A 2K6; e-mail:

    Inna Bumagina

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  1. Inna Bumagina
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Bumagina, I. The Hopf Property for Subgroups of Hyperbolic Groups. Geometriae Dedicata 106, 211–230 (2004). https://doi.org/10.1023/B:GEOM.0000033859.35966.4a

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