The conjugacy problem for relatively hyperbolic groups

Bumagin, Inna¹

¹Department of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Herzberg Building, Ottawa, Ontario, Canada K1S 5B6

Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 1013-1040

Résumé

Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810–840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups.

Détail

DOI : 10.2140/agt.2004.4.1013

Keywords: negatively curved groups, algorithmic problems

Bibliographie

[1] E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459

[2] J M Alonso, E Al., Notes on word hyperbolic groups, from: "Group theory from a geometrical viewpoint (Trieste, 1990)", World Sci. Publ., River Edge, NJ (1991) 3

[3] B Baumslag, Residually free groups, Proc. London Math. Soc. $(3)$ 17 (1967) 402

[4] B H Bowditch, Relatively hyperbolic groups, preprint, University of Southampton (1998)

[5] B H Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999) 3673

[6] B H Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999) 509

[7] B H Bowditch, Peripheral splittings of groups, Trans. Amer. Math. Soc. 353 (2001) 4057

[8] I Bumagin, On definitions of relatively hyperbolic groups, from: "Geometric methods in group theory", Contemp. Math. 372, Amer. Math. Soc. (2005) 189

[9] J W Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984) 123

[10] D J Collins, C F Miller Iii, The conjugacy problem and subgroups of finite index, Proc. London Math. Soc. $(3)$ 34 (1977) 535

[11] F Dahmani, Classifying spaces and boundaries for relatively hyperbolic groups, Proc. London Math. Soc. $(3)$ 86 (2003) 666

[12] F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933

[13] M Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911) 116

[14] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810

[15] A Juhász, Extension of group presentations and relative small cancellation theory I, Internat. J. Algebra Comput. 10 (2000) 375

[16] , Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser (1990)

[17] B Goldfarb, Novikov conjectures and relative hyperbolicity, Math. Scand. 85 (1999) 169

[18] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[19] G C Hruska, Nonpositively curved 2–complexes with isolated flats, Geom. Topol. 8 (2004) 205

[20] I Kapovich, P Schupp, Relative hyperbolicity and Artin groups, Geom. Dedicata 107 (2004) 153

[21] O Kharlampovich, A Myasnikov, Irreducible affine varieties over a free group II: Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra 200 (1998) 517

[22] O G Kharlampovich, A G Myasnikov, V N Remeslennikov, D E Serbin, Subgroups of fully residually free groups: algorithmic problems, from: "Group theory, statistics, and cryptography", Contemp. Math. 360, Amer. Math. Soc. (2004) 63

[23] I G Lysënok, Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 814, 912

[24] H A Masur, Y N Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999) 103

[25] A W Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. 59 (1966) 123

[26] A G Myasnikov, V N Remeslennikov, D E Serbin, Regular free length functions on Lyndon's free $\mathbb{Z}[t]$–group $F^{Z[t]}$, from: "Groups, languages, algorithms", Contemp. Math. 378, Amer. Math. Soc. (2005) 37

[27] D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006)

[28] D V Osin, Weak hyperbolicity and free constructions, from: "Group theory, statistics, and cryptography", Contemp. Math. 360, Amer. Math. Soc. (2004) 103

[29] D Rebbechi, Algorithmic properties of relatively hyperbolic groups, PhD thesis, Rutgers University (2000)

[30] Z Sela, Diophantine geometry over groups I: Makanin–Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) 31

[31] A Szczepański, Relatively hyperbolic groups, Michigan Math. J. 45 (1998) 611

[32] A Szczepański, Examples of relatively hyperbolic groups, Geom. Dedicata 93 (2002) 139

[33] A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41