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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**Gilbert Baumslag, Martin R. Bridson, Charles F. Miller III, and Hamish Short,*Fibre products, non-positive curvature, and decision problems*, Comment. Math. Helv.**75**(2000), no. 3, 457–477. MR**1793798**, 10.1007/s000140050136**2.**Gilbert Baumslag and James E. Roseblade,*Subgroups of direct products of free groups*, J. London Math. Soc. (2)**30**(1984), no. 1, 44–52. MR**760871**, 10.1112/jlms/s2-30.1.44**3.**M. Bestvina, M. Feighn, and M. Handel,*Laminations, trees, and irreducible automorphisms of free groups*, Geom. Funct. Anal.**7**(1997), no. 2, 215–244. MR**1445386**, 10.1007/PL00001618**4.**N. Brady and A. Miller,*CAT**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.**Fritz J. Grunewald,*On some groups which cannot be finitely presented*, J. London Math. Soc. (2)**17**(1978), no. 3, 427–436. MR**500627**, 10.1112/jlms/s2-17.3.427**9.**Roger C. Lyndon and Paul E. Schupp,*Combinatorial group theory*, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR**0577064****10.**Charles F. Miller III,*On group-theoretic decision problems and their classification*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 68. MR**0310044****11.**Charles F. Miller III,*Decision problems for groups—survey and reflections*, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 1–59. MR**1230627**, 10.1007/978-1-4613-9730-4_1**12.**Lee Mosher,*A hyperbolic-by-hyperbolic hyperbolic group*, Proc. Amer. Math. Soc.**125**(1997), no. 12, 3447–3455. MR**1443845**, 10.1090/S0002-9939-97-04249-4**13.**Elvira Strasser Rapaport,*Note on Nielsen transformations*, Proc. Amer. Math. Soc.**10**(1959), 228–235. MR**0104724**, 10.1090/S0002-9939-1959-0104724-1**14.**E. Rips,*Subgroups of small cancellation groups*, Bull. London Math. Soc.**14**(1982), no. 1, 45–47. MR**642423**, 10.1112/blms/14.1.45

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20F10,
20F67

Retrieve articles in all journals with MSC (2000): 20F10, 20F67

Additional Information

**Martin R. Bridson**

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

Email:
m.bridson@ic.ac.uk

**Charles F. Miller III**

Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville 3052, Australia

Email:
c.miller@ms.unimelb.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-03-07008-4

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