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Recognition of subgroups of direct products of hyperbolic groups

Authors: Martin R. Bridson and Charles F. Miller III
Journal: Proc. Amer. Math. Soc. 132 (2004), 59-65
MSC (2000): Primary 20F10, 20F67
Published electronically: June 5, 2003
MathSciNet review: 2021248
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Abstract: We give examples of direct products of three hyperbolic groups in which there cannot exist an algorithm to decide which finitely presented subgroups are isomorphic.

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  • 1. G. Baumslag, M. R. Bridson, C. F. Miller III and H. Short, Fibre products, non-positive curvature, and decision problems, Comment. Math. Helv. 75 (2000), 457-477. MR 2001k:20091
  • 2. G. Baumslag and J. E. Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1984), 44-52. MR 86d:20028
  • 3. M. Bestvina, M. Handel and M. Feighn, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), no. 2, 215-244. MR 98c:20045
  • 4. N. Brady and A. Miller, CAT$(-1)$ structures for free-by-free groups, Geom. Dedicata 90 (2002), 77-98.
  • 5. M. R. Bridson, Subgroups of semihyperbolic groups, Monographie de L'Enseign. Math. 38 (2001), 85-111.
  • 6. M. R. Bridson, The conjugacy and isomorphism problems for combable groups, Math. Ann., to appear.
  • 7. M. R. Bridson and C. F. Miller III, Finiteness conditions for subdirect products of groups, in preparation.
  • 8. F. Grunewald, On some groups which cannot be finitely presented, J. London Math. Soc. (2) 17 (1978), 427-436. MR 80d:20033
  • 9. R. C. Lyndon and P. E. Schupp, ``Combinatorial group theory", Springer-Verlag, Berlin, Heidelberg, New York, 1977. MR 58:28182
  • 10. C. F. Miller III, ``On group-theoretic decision problems and their classification", Annals of Mathematics Studies, No. 68, Princeton University Press (1971). MR 46:9147
  • 11. C. F. Miller III, Decision problems for groups: survey and reflections, in ``Algorithms and Classification in Combinatorial Group Theory" (eds. G. Baumslag and C. F. Miller III), MSRI Publications No. 23, Springer-Verlag (1992), 1-59. MR 94i:20057
  • 12. L. Mosher, A hyperbolic-by-hyperbolic hyperbolic group, Proc. Amer. Math. Soc. 125 (1997), 3447-3455. MR 98c:20069
  • 13. E. S. Rapaport, Note on Nielsen transformations, Proc. Amer. Math. Soc. 10 (1959), 229-235. MR 21:3477
  • 14. E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), 45-47. MR 83c:20049

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Additional Information

Martin R. Bridson
Affiliation: Department of Mathematics, 180 Queen’s Gate, London SW7 2BZ, United Kingdom

Charles F. Miller III
Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville 3052, Australia

Keywords: Hyperbolic groups, finitely presented subgroups, isomorphism problem
Received by editor(s): January 31, 2002
Received by editor(s) in revised form: September 9, 2002
Published electronically: June 5, 2003
Additional Notes: The first author’s research was funded by an Advanced Fellowship from the EPSRC
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2003 American Mathematical Society

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