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The conjugacy problem for relatively hyperbolic groups. (English) Zbl 1111.20035

The relatively hyperbolic groups introduced by M. Gromov [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] are coarsely negatively curved relative to certain subgroups, called parabolic subgroups. The motivating examples are fundamental groups of negatively curved manifolds with cusps that are hyperbolic relative to the fundamental groups of the cusps. B. Farb gave his own definition of relatively hyperbolic groups, using Cayley graphs [Geom. Funct. Anal. 8, No. 5, 810-840 (1998; Zbl 0985.20027)]. In central position in Farb’s considerations is the Bounded Coset Property (BCP). The author proves that hyperbolic group \(G\) relative to a subgroup \(H\) has solvable conjugacy problem, provided that it is solvable in \(H\). The author mentions the importance of the BCP property for solvability of the conjugacy problem. Some applications to fundamental groups and limit groups are given.

MSC:

20F67 Hyperbolic groups and nonpositively curved groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F65 Geometric group theory
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53A35 Non-Euclidean differential geometry
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References:

[1] E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459 · Zbl 1074.57001 · doi:10.1112/S0024609304004059
[2] J M Alonso, e al., Notes on word hyperbolic groups, World Sci. Publ., River Edge, NJ (1991) 3 · Zbl 0849.20023
[3] B Baumslag, Residually free groups, Proc. London Math. Soc. \((3)\) 17 (1967) 402 · Zbl 0166.01502 · doi:10.1112/plms/s3-17.3.402
[4] B H Bowditch, Relatively hyperbolic groups, preprint, University of Southampton (1998) · Zbl 0918.20027
[5] B H Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999) 3673 · Zbl 0938.20033 · doi:10.1090/S0002-9947-99-02388-0
[6] B H Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999) 509 · Zbl 0926.20027 · doi:10.1007/PL00004703
[7] B H Bowditch, Peripheral splittings of groups, Trans. Amer. Math. Soc. 353 (2001) 4057 · Zbl 1037.20041 · doi:10.1090/S0002-9947-01-02835-5
[8] I Bumagin, On definitions of relatively hyperbolic groups, Contemp. Math. 372, Amer. Math. Soc. (2005) 189 · Zbl 1091.20029
[9] J W Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984) 123 · Zbl 0606.57003 · doi:10.1007/BF00146825
[10] D J Collins, C F Miller III, The conjugacy problem and subgroups of finite index, Proc. London Math. Soc. \((3)\) 34 (1977) 535 · Zbl 0364.20043 · doi:10.1112/plms/s3-34.3.535
[11] F Dahmani, Classifying spaces and boundaries for relatively hyperbolic groups, Proc. London Math. Soc. \((3)\) 86 (2003) 666 · Zbl 1031.20039 · doi:10.1112/S0024611502013989
[12] F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933 · Zbl 1037.20042 · doi:10.2140/gt.2003.7.933
[13] M Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911) 116 · JFM 42.0508.03 · doi:10.1007/BF01456932
[14] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 · Zbl 0985.20027 · doi:10.1007/s000390050075
[15] A Juhász, Extension of group presentations and relative small cancellation theory I, Internat. J. Algebra Comput. 10 (2000) 375 · Zbl 1015.20025 · doi:10.1142/S0218196700000157
[16] , Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser (1990) · Zbl 0731.20025
[17] B Goldfarb, Novikov conjectures and relative hyperbolicity, Math. Scand. 85 (1999) 169 · Zbl 0981.19002
[18] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015
[19] G C Hruska, Nonpositively curved 2-complexes with isolated flats, Geom. Topol. 8 (2004) 205 · Zbl 1063.20048 · doi:10.2140/gt.2004.8.205
[20] I Kapovich, P Schupp, Relative hyperbolicity and Artin groups, Geom. Dedicata 107 (2004) 153 · Zbl 1080.20032 · doi:10.1007/s10711-004-9285-0
[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 · Zbl 0904.20017 · doi:10.1006/jabr.1997.7184
[22] O G Kharlampovich, A G Myasnikov, V N Remeslennikov, D E Serbin, Subgroups of fully residually free groups: algorithmic problems, Contemp. Math. 360, Amer. Math. Soc. (2004) 63 · Zbl 1103.20029
[23] I G Lysënok, Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 814, 912 · Zbl 0692.20022
[24] H A Masur, Y N Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999) 103 · Zbl 0941.32012 · doi:10.1007/s002220050343
[25] A W Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. 59 (1966) 123 · Zbl 0143.03701
[26] A G Myasnikov, V N Remeslennikov, D E Serbin, Regular free length functions on Lyndon’s free \(\mathbbZ[t]\)-group \(F^{Z[t]}\), Contemp. Math. 378, Amer. Math. Soc. (2005) 37 · Zbl 1137.20026
[27] D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006) · Zbl 1093.20025
[28] D V Osin, Weak hyperbolicity and free constructions, Contemp. Math. 360, Amer. Math. Soc. (2004) 103 · Zbl 1074.20029
[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 · Zbl 1018.20034 · doi:10.1007/s10240-001-8188-y
[31] A Szczepański, Relatively hyperbolic groups, Michigan Math. J. 45 (1998) 611 · Zbl 0962.20031 · doi:10.1307/mmj/1030132303
[32] A Szczepański, Examples of relatively hyperbolic groups, Geom. Dedicata 93 (2002) 139 · Zbl 1046.20031 · doi:10.1023/A:1020380603501
[33] A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41 · Zbl 1043.20020 · doi:10.1515/crll.2004.007
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