Mathematische Annalen

, Volume 331, Issue 1, pp 1–19

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

Article

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)MATHGoogle Scholar
  2. 2.
    Arzhantseva, G.: Generic properties of finitely presented groups and Howson’s theorem. Comm. Algebra 26 (11), 3783–3792 (1998)MATHGoogle Scholar
  3. 3.
    Arzhantseva, G.: A property of subgroups of infinite index in a free group. Proc. Amer. Math. Soc. 128 (11), 3205–3210 (2000)CrossRefMATHGoogle Scholar
  4. 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. 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)MATHGoogle Scholar
  6. 6.
    Brinkmann, P.: Perimeter and coherence according to McCammond and Wise. Groups—Korea ‘98 (Pusan), de Gruyter, Berlin, 2000, pp. 81–90Google Scholar
  7. 7.
    Brunner, A.: A group with an infinite number of Nielsen inequivalent one-relator presentations. J. Algebra 42 (1), 81–84 (1976)MATHGoogle Scholar
  8. 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)CrossRefMATHGoogle Scholar
  9. 9.
    Champetier, C.: Petite simplification dans les groupes hyperboliques. Ann. Fac. Sci. Toulouse math. (6) 3 (2), 161–221 (1994)Google Scholar
  10. 10.
    Champetier, C.: Propriétés statistiques des groupes de présentation finie. Adv. Math. 116, 197–262 (1995)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Champetier, C.: The space of finitely generated groups. Topology 39, 657–680 (2000)CrossRefMathSciNetMATHGoogle Scholar
  12. 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)MATHGoogle Scholar
  13. 13.
    Cherix, P.-A., Schaeffer, G.: An asymptotic Freiheitssatz for finitely generated groups. Enseign. Math. (2) 44, 9–22 (1998)Google Scholar
  14. 14.
    Collins, D.: Presentations of the amalgamated free product of two infinite cycles. Math. Ann. 237 (3), 233–241 (1978)MATHGoogle Scholar
  15. 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. 177–192Google Scholar
  16. 16.
    Epstein, D., Cannon, J., Holt, D., Levy, S., Paterson, M., Thurston, W.: Word Processing in Groups. Jones and Bartlett, Boston, 1992Google Scholar
  17. 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. 13–36Google Scholar
  18. 18.
    Fine, B., Rosenberger, G., Stille, M.: Nielsen transformations and applications: a survey. Groups—Korea ‘94 (Pusan), de Gruyter, Berlin, 1995, pp. 69–105Google Scholar
  19. 19.
    Gromov, M.: Hyperbolic Groups. In: “Essays in Group Theory (G.M.Gersten, editor)”. MSRI publ. 8, 75–263 (1987)Google Scholar
  20. 20.
    Gromov, M.: Random walks in random groups. Geom. Funct. Analysis 13 (1), 73–146 (2003)MATHGoogle Scholar
  21. 21.
    Ghys, E., de la Harpe, P. (eds.): Sur les groupes hyperboliques d’aprés Mikhael Gromov. Birkhäuser, Progress in Mathematics series, vol. 83, (1990)Google Scholar
  22. 22.
    Hill, P., Pride, S., Vella, A.: Subgroups of small cancellation groups. J. Reine Angew. Math. 349, 24–54 (1984)MathSciNetMATHGoogle Scholar
  23. 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)CrossRefMATHGoogle Scholar
  24. 24.
    Kapovich, I., Myasnikov, A.: Stallings foldings and the subgroup structure of free groups. J. Algebra 248 (2), 608–668 (2002)CrossRefMATHGoogle Scholar
  25. 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)CrossRefMATHGoogle Scholar
  26. 26.
    Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Average-case complexity and decision problems in group theory. Advances in Mathematics (to appear)Google Scholar
  27. 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)CrossRefMATHGoogle Scholar
  28. 28.
    Kapovich, I., Schupp, P., Shpilrain, V.: Generic properties of Whitehead’s Algorithm, stabilizers in Aut(Fk) and one-relator groups. Preprint, 2003, http://front.math.ucdavis.edu/math.GR/0303386
  29. 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)MathSciNetMATHGoogle Scholar
  30. 30.
    Kapovich, I., Weidmann, R.: Nielsen methods and groups acting on hyperbolic spaces. Geom. Dedicata 98 (1), 95–121 (2003)CrossRefMATHGoogle Scholar
  31. 31.
    Kapovich, I., Weidmann, R.: Freely indecomposable groups acting on hyperbolic spaces. Intern. J. Algebra Comput. 14 (2), 115–172 (2004)CrossRefGoogle Scholar
  32. 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. 33.
    Magnus, W.: Uber freie Faktorgruppen und freie Untergruppen gegebener Gruppen. Monatsch. Math. Phys. 47, 307–313 (1939)MATHGoogle Scholar
  34. 34.
    Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. Wiley, New York, 1966Google Scholar
  35. 35.
    Lyndon, R., Schupp, P.: Combinatorial Group Theory, Springer-Verlag, 1977. Reprinted in the “Classics in mathematics” series, 2000Google Scholar
  36. 36.
    McCammond, J., Wise, D.: Coherence, local quasiconvexity and the perimeter of 2-complexes. PreprintGoogle Scholar
  37. 37.
    McCool, J., Pietrowski, A.: On free products with amalgamation of two infinite cyclic groups. J. Algebra 18, 377–383 (1971)MATHGoogle Scholar
  38. 38.
    Ollivier, Y.: Critical densities for random quotients of hyperbolic groups. C. R. Math. Acad. Sci. Paris 336 (5), 391–394 (2003)MATHGoogle Scholar
  39. 39.
    Ol’shanskii, A.Yu.: Almost every group is hyperbolic. Internat. J. Algebra Comput. 2, 1–17 (1992)MathSciNetGoogle Scholar
  40. 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, 1996Google Scholar
  41. 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. 42.
    Pride, S.: The two-generator subgroups of one-relator groups with torsion. Trans. Amer. Math. Soc. 234 (2), 483–496 (1977)MATHGoogle Scholar
  43. 43.
    Pride, S.: The isomorphism problem for two-generator one-relator groups with torsion is solvable. Trans. Amer. Math. Soc. 227, 109–139 (1977)MATHGoogle Scholar
  44. 44.
    Pride, S.: Small cancellation conditions satisfied by one-relator groups. Math. Z. 184 (2), 283–286 (1983)MATHGoogle Scholar
  45. 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, 1982Google Scholar
  46. 46.
    Rosenberger, G.: The isomorphism problem for cyclically pinched one-relator groups. J. Pure Appl. Algebra 95 (1), 75–86 (1994)CrossRefMATHGoogle Scholar
  47. 47.
    Schupp, P.E.: Coxeter Groups, 2-Completion, Perimeter Reduction and Subgroup Separability. Geom. Dedicata 96, 179–198 (2003)CrossRefMathSciNetGoogle Scholar
  48. 48.
    Sela, Z.: The isomorphism problem for hyperbolic groups. I. Ann. of Math. (2) 141 (2), 217–283 (1995)Google Scholar
  49. 49.
    Stallings, J.R.: Topology of finite graphs. Invent. Math. 71 (3), 551–565 (1983)MATHGoogle Scholar
  50. 50.
    Vershik, A.: Dynamic theory of growth in groups: entropy, boundaries, examples. Uspekhi Mat. Nauk 55 (4(334)), 59–128 (2000)MATHGoogle Scholar
  51. 51.
    White, M.: Injectivity radius and fundamental groups of hyperbolic 3-manifolds. Comm. Anal. Geom. 10 (2), 377–395 (2002)MATHGoogle Scholar
  52. 52.
    Whitehead, J.H.C.: On equivalent sets of elements in free groups. Ann. Math. 37, 782–800 (1936)MATHGoogle Scholar
  53. 53.
    Wise, D.: The residual finiteness of negatively curved polygons of finite groups. Invent. Math. 149 (3), 579–617 (2002)CrossRefMATHGoogle Scholar
  54. 54.
    Zieschang, H.: Generators of the free product with amalgamation of two infinite cyclic groups. Math. Ann. 227 (3), 195–221 (1977)MATHGoogle Scholar
  55. 55.
    Zuk, A.: On property (T) for discrete groups. Rigidity and Dynamics in Geometry (M. Burger and A. Iozzi, editors), 469–482, Springer, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations