Skip to main content
Log in

The isomorphism problem for toral relatively hyperbolic groups

Publications mathématiques Aims and scope Submit manuscript

Abstract

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. I. Adian, The unsolvability of certain algorithmic problems in the theory of groups, Trudy Moskov. Obsc., 6 (1957), 231–298.

    Google Scholar 

  2. E. Alibegović, A combination theorem for relatively hyperbolic groups, Bull. Lond. Math. Soc., 37 (2005), 459–466.

    Article  MATH  Google Scholar 

  3. G. Baumslag, D. Gildenhuys and R. Strebel, Algorithmically insoluble problems about finitely presented soluble groups, Lie and associative algebras, I, J. Pure Appl. Algebra, 39 (1986), 53–94.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Belegradek, Aspherical manifolds with relatively hyperbolic fundamental groups, Geom. Dedicata, 129 (2007), 119–144.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Bestvina, Degenerations of hyperbolic space, Duke Math. J., 56 (1988), 143–161.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Bestvina and M. Feighn, Bounding the complexity of simplicial group actions, Invent. Math., 103 (1991), 449–469.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. B. H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math., 180 (1998), 145–186.

    Article  MATH  MathSciNet  Google Scholar 

  9. B. H. Bowditch, Relatively Hyperbolic Groups, preprint (1999).

  10. B. H. Bowditch, Peripheral splittings of groups, Trans. Amer. Math. Soc., 353 (2001), 4057–4082.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  12. K. S. Brown, Cohomology of Groups, Grad. Texts Math., vol. 87, Springer, New York, Berlin, 1982.

    MATH  Google Scholar 

  13. I. Bumagin, The conjugacy problem for relatively hyperbolic groups, Algebr. Geom. Topol., 4 (2004), 1013–1040.

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Bumagin, O. Kharlampovich, and A. Miasnikov, Isomorphism problem for finitely generated fully residually free groups, J. Pure Appl. Algebra, 208 (2007), 961–977.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Les groupes hyperboliques de M. Gromov, Lect. Notes Math., vol. 1441, Springer, Berlin, 1991.

  16. F. Dahmani, Les groupes relativement hyperboliqes et leurs bords, PhD Thesis, Strasbourg (2003)

  17. F. Dahmani, Combination of convergence groups, Geom. Topol., 7 (2003), 933–963.

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Dahmani, Finding relative hyperbolic structures, Bull. Lond. Math. Soc., 40 (2008), 395–404.

    Article  MATH  MathSciNet  Google Scholar 

  19. F. Dahmani, Existential questions in (relatively) hyperbolic groups, Isr. J. Math., to appear.

  20. F. Dahmani and D. Groves, Detecting free splittings in relatively hyperbolic groups, Trans. Amer. Math. Soc., to appear.

  21. M. Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann., 71 (1912), 413–421.

    Article  MathSciNet  Google Scholar 

  22. M. Dehn, Papers on Group Theory and Topology, Translated from the German and with Introductions and an Appendix by John Stillwell, with an Appendix by Otto Schreier, Springer, New York, 1987.

    Google Scholar 

  23. V. Diekert, C. Gutiérrez, and C. Hagenah, The existential theory of equations with rational constraints in free groups is PSPACE-complete, in Inform. Comput., 202 (2005), 105–140.

    Article  MATH  Google Scholar 

  24. V. Diekert and A. Muscholl, Solvability of equations in free partially commutative groups is decidable, in Automata, Languages and Programming, Lect. Notes Comput. Sci., vol. 2076, Springer, 2001, pp. 543–554.

  25. C. Druţu and M. Sapir, Tree-graded spaces and asymptotic cones of groups, Topology, 44 (2005), 959–1058.

    Article  MATH  MathSciNet  Google Scholar 

  26. C. Druţu and M. Sapir, Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups, Adv. Math., 217 (2007), 1313–1367.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, and W. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992.

    MATH  Google Scholar 

  29. B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal., 8 (1998), 810–840.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Forester, On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv., 78 (2003), 740–751.

    Article  MATH  MathSciNet  Google Scholar 

  31. K. Fujiwara and P. Papasoglu, JSJ decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal., 16 (2006), 70–125.

    Article  MATH  MathSciNet  Google Scholar 

  32. V. Gerasimov, Detecting connectedness of the boundary of a hyperbolic group, preprint.

  33. R. Grigorchuk and I. Lysenok, A description of solutions of quadratic equations in hyperbolic groups, Int. J. Algebra Comput., 2 (1992), 237–274.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263.

  35. D. Groves, Limits of certain CAT(0) groups, I: Compactification, Algebr. Geom. Topol., 5 (2005), 1325–1364.

    Article  MATH  MathSciNet  Google Scholar 

  36. D. Groves, Limit groups for relatively hyperbolic groups, I: The basic tools, preprint.

  37. D. Groves, Limit groups for relatively hyperbolic groups, II: Makanin–Razborov diagrams, Geom. Topol., 9 (2005), 2319–2358.

    Article  MATH  MathSciNet  Google Scholar 

  38. F. Grunewald and D. Segal, Some general algorithms. II. Nilpotent groups, Ann. Math. (2), 112 (1980), 585–617.

    Article  MathSciNet  Google Scholar 

  39. V. Guirardel, Limit groups and groups acting freely on R n-trees, Geom. Topol., 8 (2004), 1427–1470.

    Article  MATH  MathSciNet  Google Scholar 

  40. G. C. Hruska and B. Kleiner, Hadamard spaces with isolated flats, with an appendix by the authors and Mohamad Hindawi, Geom. Topol., 9 (2005), 1501–1538.

    Article  MATH  MathSciNet  Google Scholar 

  41. C. Hummel, Rank one lattices whose parabolic isometries have no rational part, Proc. Amer. Math. Soc., 126 (1998), 2453–2458.

    Article  MATH  MathSciNet  Google Scholar 

  42. O. Kharlampovich and A. Miasnikov, Effective JSJ decompositions, in Groups, Languages and Algorithms, Contemp. Math., vol. 378, Amer. Math. Soc., Providence, RI, 2005, pp. 87–212.

    Google Scholar 

  43. O. Kharlampovich and A. Miasnikov, Elementary theory of free non-abelian groups, J. Algebra, 302 (2006), 451–552.

    Article  MATH  MathSciNet  Google Scholar 

  44. G. Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata, 114 (2005), 49–70.

    Article  MATH  MathSciNet  Google Scholar 

  45. G. Levitt, Characterizing rigid simplicial actions on trees, in Geometric Methods in Group Theory, Contemp. Math., vol. 372, Amer. Math. Soc., Providence, RI, 2005, pp. 27–33.

    Google Scholar 

  46. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergeb. Math. Grenzgeb., vol. 89, Springer, Berlin, 1977.

    MATH  Google Scholar 

  47. I. Lysenok, Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat., 53 (1989), 814–832, 912; transl. in Math. USSR-Izv., 35 (1990), 145–163.

  48. G. S. Makanin, Decidability of the universal and positive theories of a free group (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 735–749; transl. in Math. USSR-Izv., 25 (1985), 75–88.

  49. C. F. Miller III, On Group Theoretic Decision Problems and Their Classification, Ann. Math. Stud., vol. 68, Princeton University Press, Princeton, 1971.

    MATH  Google Scholar 

  50. C. F. Miller III, Decision problems for groups – survey and reflections, in Algorithms and Classification in Combinatorial Group Theory (Berkeley, CA, 1989), MSRI Publ., vol. 23, Springer, New York, 1992, pp. 1–59.

    Google Scholar 

  51. D. V. Osin, Relatively Hyperbolic Groups: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems, Mem. Amer. Math. Soc., vol. 179, Amer. Math. Soc., Providence, RI, 2006.

    Google Scholar 

  52. P. Papasoglu, An algorithm detecting hyperbolicity, in Geometric and Computational Perspectives on Infinite Groups (Minneapolis, MN and New Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 25, Amer. Math. Soc., Providence, RI, 1996, pp. 193–200.

  53. F. Paulin, Topologie de Gromov équivariante, structures hyperbolique et arbres réels, Invent. Math., 94 (1988), 53–80.

    Article  MATH  MathSciNet  Google Scholar 

  54. M. O. Rabin, Recursive unsolvability of group theoretic problems, Ann. Math., 67 (1958), 172–194.

    Article  MathSciNet  Google Scholar 

  55. D. Rebbechi, Algorithmic Properties of Relatively Hyperbolic Groups, PhD Thesis, ArXiv math.GR/0302245.

  56. E. Rips and Z. Sela, Canonical representatives and equations in hyperbolic groups, Invent. Math., 120 (1995), 489–512.

    Article  MATH  MathSciNet  Google Scholar 

  57. E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. Math. (2), 146 (1997), 53–104.

    Article  MathSciNet  Google Scholar 

  58. V. Roman’kov, Universal theory of nilpotent groups, Mat. Zametki, 25 (1979), 487–495, 635.

  59. D. Segal, Decidable properties of polycyclic groups, Proc. Lond. Math. Soc., III. Ser., 61 (1990), 497–528.

    Article  MATH  Google Scholar 

  60. Z. Sela, The isomorphism problem for hyperbolic groups I, Ann. Math. (2), 141 (1995), 217–283.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  62. Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, Geom. Funct. Anal., 7 (1997), 561–593.

    Article  MATH  MathSciNet  Google Scholar 

  63. Z. Sela, Diophantine geometry over groups, I: Makanin–Razborov diagrams, Publ. Math., Inst. Hautes Étud. Sci., 93 (2003), 31–105.

    Google Scholar 

  64. Z. Sela, Diophantine geometry over groups, VI: The elementary theory of a free group, Geom. Funct. Anal., 16 (2006), 707–730.

    MATH  MathSciNet  Google Scholar 

  65. Z. Sela, Diophantine Geometry over Groups, VIII: Elementary Theory of Hyperbolic Groups, preprint (2002).

  66. J.-P. Serre, Trees, Springer, Berlin, 1980.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to François Dahmani.

About this article

Cite this article

Dahmani, F., Groves, D. The isomorphism problem for toral relatively hyperbolic groups. Publ.math.IHES 107, 211–290 (2008). https://doi.org/10.1007/s10240-008-0014-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10240-008-0014-3

Keywords

Navigation