Skip to main content
SpringerLink
Log in

Genericity, the Arzhantseva-Ol’shanskii method and the isomorphism problem for one-relator groups

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

We show that the isomorphism problem is solvable in at most exponential time for a class of one-relator groups which is exponentially generic in the sense of Ol’shanskii. This is obtained by applying the Arzhantseva-Ol’shanskii graph minimization method to prove the general result that for fixed integers m≥2 and n≥1 there is an exponentially generic class of non-free m-generator n-relator groups with the property that there is only one Nielsen equivalence class of m-tuples which generate a non-free subgroup. In particular, every m-generated subgroup in such a generic group G is either free or is equal to G itself and such groups are thus co-Hopfian. These results are obtained by elementary methods without using the deep results of Sela about co-Hopficity and the isomorphism problem for torsion-free hyperbolic groups.

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

Access this article

Log in via an institution

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. Arzhantseva, G.: On groups in which subgroups with a fixed number of generators are free, (Russian) Fundam. Prikl. mat. 3 (3), 675–683 (1997)

    MATH  Google Scholar 

  2. Arzhantseva, G.: Generic properties of finitely presented groups and Howson’s theorem. Comm. Algebra 26 (11), 3783–3792 (1998)

    MATH  Google Scholar 

  3. Arzhantseva, G.: A property of subgroups of infinite index in a free group. Proc. Amer. Math. Soc. 128 (11), 3205–3210 (2000)

    Article  MATH  Google Scholar 

  4. Arzhantseva, G.: A dichotomy for finitely generated subgroups of word-hyperbolic groups, preprint, University of Geneve, 2001; http://www.unige.ch/math/biblio/preprint/pp2001.html

  5. Arzhantseva, G., Ol’shanskii, A.: Genericity of the class of groups in which subgroups with a lesser number of generators are free, (Russian). Mat. Zametki 59 (4), 489–496 (1996)

    MATH  Google Scholar 

  6. Brinkmann, P.: Perimeter and coherence according to McCammond and Wise. Groups—Korea ‘98 (Pusan), de Gruyter, Berlin, 2000, pp. 81–90

  7. Brunner, A.: A group with an infinite number of Nielsen inequivalent one-relator presentations. J. Algebra 42 (1), 81–84 (1976)

    MATH  Google Scholar 

  8. Bumagin, I.: On Small Cancellation k-Generated Groups with (k -1)-Generated Subgroups All Free. Intern J. Alg. Comp. 11(5), 507–524 (2001)

    Article  MATH  Google Scholar 

  9. Champetier, C.: Petite simplification dans les groupes hyperboliques. Ann. Fac. Sci. Toulouse math. (6) 3 (2), 161–221 (1994)

    Google Scholar 

  10. Champetier, C.: Propriétés statistiques des groupes de présentation finie. Adv. Math. 116, 197–262 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Champetier, C.: The space of finitely generated groups. Topology 39, 657–680 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cherix, P.-A., Valette, A.: On spectra of simple random walks on one-relator groups. With an appendix by Paul Jolissaint. Pacific J. Math. 175, 417–438 (1996)

    MATH  Google Scholar 

  13. Cherix, P.-A., Schaeffer, G.: An asymptotic Freiheitssatz for finitely generated groups. Enseign. Math. (2) 44, 9–22 (1998)

    Google Scholar 

  14. Collins, D.: Presentations of the amalgamated free product of two infinite cycles. Math. Ann. 237 (3), 233–241 (1978)

    MATH  Google Scholar 

  15. Delzant, T.: Sous-groupes a deux generateurs des groups hyperboliques. In: “Group theory from a geometric viewpoint”, Proc. ICTP. Trieste, World Scientific, Singapore, 1991, pp. 177–192

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

  17. Fine, B., Röhl, F., Rosenberger, G.: On HNN-groups whose three-generator subgroups are free. Infinite groups and group rings (Tuscaloosa, AL, 1992), Ser. Algebra, 1, World Sci. Publishing, River Edge, NJ, 1993, pp. 13–36

  18. Fine, B., Rosenberger, G., Stille, M.: Nielsen transformations and applications: a survey. Groups—Korea ‘94 (Pusan), de Gruyter, Berlin, 1995, pp. 69–105

  19. Gromov, M.: Hyperbolic Groups. In: “Essays in Group Theory (G.M.Gersten, editor)”. MSRI publ. 8, 75–263 (1987)

    Google Scholar 

  20. Gromov, M.: Random walks in random groups. Geom. Funct. Analysis 13 (1), 73–146 (2003)

    MATH  Google Scholar 

  21. Ghys, E., de la Harpe, P. (eds.): Sur les groupes hyperboliques d’aprés Mikhael Gromov. Birkhäuser, Progress in Mathematics series, vol. 83, (1990)

  22. Hill, P., Pride, S., Vella, A.: Subgroups of small cancellation groups. J. Reine Angew. Math. 349, 24–54 (1984)

    MathSciNet  MATH  Google Scholar 

  23. Ivanov, S.V., Schupp, P.E.: On the hyperbolicity of small cancellation groups and one-relator groups. Trans. Amer. Math. Soc. 350 (5), 1851–1894 (1998)

    Article  MATH  Google Scholar 

  24. Kapovich, I., Myasnikov, A.: Stallings foldings and the subgroup structure of free groups. J. Algebra 248 (2), 608–668 (2002)

    Article  MATH  Google Scholar 

  25. Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory and random walks. J. Algebra 264 (2), 665–694 (2003)

    Article  MATH  Google Scholar 

  26. Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Average-case complexity and decision problems in group theory. Advances in Mathematics (to appear)

  27. Kapovich, I., Schupp, P.: Bounded rank subgroups of Coxeter groups, Artin groups and One-relator groups with torsion. Proceedings of the London Math. Soc. 88 (1), 89–113 (2004)

    Article  MATH  Google Scholar 

  28. Kapovich, I., Schupp, P., Shpilrain, V.: Generic properties of Whitehead’s Algorithm, stabilizers in Aut(F k ) and one-relator groups. Preprint, 2003, http://front.math.ucdavis.edu/math.GR/0303386

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

    MathSciNet  MATH  Google Scholar 

  30. Kapovich, I., Weidmann, R.: Nielsen methods and groups acting on hyperbolic spaces. Geom. Dedicata 98 (1), 95–121 (2003)

    Article  MATH  Google Scholar 

  31. Kapovich, I., Weidmann, R.: Freely indecomposable groups acting on hyperbolic spaces. Intern. J. Algebra Comput. 14 (2), 115–172 (2004)

    Article  Google Scholar 

  32. Karrass, A., Magnus, W., Solitar, D.: Elements of finite order in groups with a single defining relation. Comm. Pure Appl. Math. 13, 57–66 (1960)

    Google Scholar 

  33. Magnus, W.: Uber freie Faktorgruppen und freie Untergruppen gegebener Gruppen. Monatsch. Math. Phys. 47, 307–313 (1939)

    MATH  Google Scholar 

  34. Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. Wiley, New York, 1966

  35. Lyndon, R., Schupp, P.: Combinatorial Group Theory, Springer-Verlag, 1977. Reprinted in the “Classics in mathematics” series, 2000

  36. McCammond, J., Wise, D.: Coherence, local quasiconvexity and the perimeter of 2-complexes. Preprint

  37. McCool, J., Pietrowski, A.: On free products with amalgamation of two infinite cyclic groups. J. Algebra 18, 377–383 (1971)

    MATH  Google Scholar 

  38. Ollivier, Y.: Critical densities for random quotients of hyperbolic groups. C. R. Math. Acad. Sci. Paris 336 (5), 391–394 (2003)

    MATH  Google Scholar 

  39. Ol’shanskii, A.Yu.: Almost every group is hyperbolic. Internat. J. Algebra Comput. 2, 1–17 (1992)

    MathSciNet  Google Scholar 

  40. Papasoglu, P.: An algorithm detecting hyperbolicity. Geometric and computational perspectives on infinite groups (Minneapolis and New Brunswick, NJ, 1994), 193–200, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 25, Amer. Math. Soc., Providence, RI, 1996

  41. Pride, S.: On the Hopficity and related properties of small cancellation groups. J. London Math. Soc. (2) 14 (2), 269–276 (1976)

    Google Scholar 

  42. Pride, S.: The two-generator subgroups of one-relator groups with torsion. Trans. Amer. Math. Soc. 234 (2), 483–496 (1977)

    MATH  Google Scholar 

  43. Pride, S.: The isomorphism problem for two-generator one-relator groups with torsion is solvable. Trans. Amer. Math. Soc. 227, 109–139 (1977)

    MATH  Google Scholar 

  44. Pride, S.: Small cancellation conditions satisfied by one-relator groups. Math. Z. 184 (2), 283–286 (1983)

    MATH  Google Scholar 

  45. Rosenberger, G.: On one-relator groups that are free products of two free groups with cyclic amalgamation. Groups—St. Andrews 1981 (St. Andrews, 1981), pp. 328–344, London Math. Soc. Lecture Note Ser., 71, Cambridge Univ. Press, Cambridge-New York, 1982

  46. Rosenberger, G.: The isomorphism problem for cyclically pinched one-relator groups. J. Pure Appl. Algebra 95 (1), 75–86 (1994)

    Article  MATH  Google Scholar 

  47. Schupp, P.E.: Coxeter Groups, 2-Completion, Perimeter Reduction and Subgroup Separability. Geom. Dedicata 96, 179–198 (2003)

    Article  MathSciNet  Google Scholar 

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

  49. Stallings, J.R.: Topology of finite graphs. Invent. Math. 71 (3), 551–565 (1983)

    MATH  Google Scholar 

  50. Vershik, A.: Dynamic theory of growth in groups: entropy, boundaries, examples. Uspekhi Mat. Nauk 55 (4(334)), 59–128 (2000)

    MATH  Google Scholar 

  51. White, M.: Injectivity radius and fundamental groups of hyperbolic 3-manifolds. Comm. Anal. Geom. 10 (2), 377–395 (2002)

    MATH  Google Scholar 

  52. Whitehead, J.H.C.: On equivalent sets of elements in free groups. Ann. Math. 37, 782–800 (1936)

    MATH  Google Scholar 

  53. Wise, D.: The residual finiteness of negatively curved polygons of finite groups. Invent. Math. 149 (3), 579–617 (2002)

    Article  MATH  Google Scholar 

  54. Zieschang, H.: Generators of the free product with amalgamation of two infinite cyclic groups. Math. Ann. 227 (3), 195–221 (1977)

    MATH  Google Scholar 

  55. Zuk, A.: On property (T) for discrete groups. Rigidity and Dynamics in Geometry (M. Burger and A. Iozzi, editors), 469–482, Springer, 2002

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL, 61801, USA

    Ilya Kapovich & Paul Schupp

Authors
  1. Ilya Kapovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Schupp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilya Kapovich.

Additional information

Mathematics Subject Classification (2000): 20F

The first author acknowledges the support of the U.S.-Israel Binational Science Foundation grant no. 1999298 and of the UIUC Research Board grants.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kapovich, I., Schupp, P. Genericity, the Arzhantseva-Ol’shanskii method and the isomorphism problem for one-relator groups. Math. Ann. 331, 1–19 (2005). https://doi.org/10.1007/s00208-004-0570-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-004-0570-x

Keywords

Search

Navigation