The isomorphism problem for toral relatively hyperbolic groups

Dahmani, François; Groves, Daniel

Publications Mathématiques de l'IHÉS, Tome 107 (2008), pp. 211-290.

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic $n$-manifolds, for $n\ge 3$. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.

[1] Adian, S.I. (1957) The unsolvability of certain algorithmic problems in the theory of groups. Trudy Moskov. Obsc. 6: pp. 231-298 | MR

[2] Alibegović, E. (2005) A combination theorem for relatively hyperbolic groups. Bull. Lond. Math. Soc. 37: pp. 459-466 | MR | Zbl

[3] Baumslag, G., Gildenhuys, D., Strebel, R. (1986) Algorithmically insoluble problems about finitely presented soluble groups, Lie and associative algebras, I. J. Pure Appl. Algebra 39: pp. 53-94 | MR | Zbl

[4] Belegradek, I. (2007) Aspherical manifolds with relatively hyperbolic fundamental groups. Geom. Dedicata 129: pp. 119-144 | MR | Zbl

[5] Bestvina, M. (1988) Degenerations of hyperbolic space. Duke Math. J. 56: pp. 143-161 | MR | Zbl

[6] Bestvina, M., Feighn, M. (1991) Bounding the complexity of simplicial group actions. Invent. Math. 103: pp. 449-469 | MR | Zbl

[7] Bestvina, M., Feighn, M. (1995) Stable actions of groups on real trees. Invent. Math. 121: pp. 287-321 | MR | Zbl

[8] Bowditch, B.H. (1998) Cut points and canonical splittings of hyperbolic groups. Acta Math. 180: pp. 145-186 | MR | Zbl

[9] B. H. Bowditch, Relatively Hyperbolic Groups, preprint (1999). | MR | Zbl

[10] Bowditch, B.H. (2001) Peripheral splittings of groups. Trans. Amer. Math. Soc. 353: pp. 4057-4082 | MR | Zbl

[11] Bridson, M.R., Swarup, G.A. (1994) On Hausdorff-Gromov convergence and a theorem of Paulin. Enseign. Math. 40: pp. 267-289 | MR | Zbl

[12] Brown, K.S. (1982) Cohomology of Groups. Springer, New York, Berlin | MR | Zbl

[13] Bumagin, I. (2004) The conjugacy problem for relatively hyperbolic groups. Algebr. Geom. Topol. 4: pp. 1013-1040 | MR | Zbl

[14] Bumagin, I., Kharlampovich, O., Miasnikov, A. (2007) Isomorphism problem for finitely generated fully residually free groups. J. Pure Appl. Algebra 208: pp. 961-977 | MR | Zbl

[15] M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Les groupes hyperboliques de M. Gromov, Lect. Notes Math., vol. 1441, Springer, Berlin, 1991. | Zbl

[16] F. Dahmani, Les groupes relativement hyperboliques et leurs bords, PhD Thesis, Strasbourg (2003) | MR

[17] Dahmani, F. (2003) Combination of convergence groups. Geom. Topol. 7: pp. 933-963 | MR | Zbl

[18] Dahmani, F. (2008) Finding relative hyperbolic structures. Bull. Lond. Math. Soc. 40: pp. 395-404 | MR | Zbl

[19] F. Dahmani, Existential questions in (relatively) hyperbolic groups, Isr. J. Math., to appear. | MR | Zbl

[20] F. Dahmani and D. Groves, Detecting free splittings in relatively hyperbolic groups, Trans. Amer. Math. Soc., to appear. | MR | Zbl

[21] Dehn, M. (1912) Über unendliche diskontinuierliche Gruppen. Math. Ann. 71: pp. 413-421 | JFM | MR

[22] Dehn, M. (1987) Papers on Group Theory and Topology, Translated from the German and with Introductions and an Appendix by John Stillwell, with an Appendix by Otto Schreier. Springer, New York | MR

[23] Diekert, V., Gutiérrez, C., Hagenah, C. (2005) The existential theory of equations with rational constraints in free groups is PSPACE-complete. Inform. Comput. 202: pp. 105-140 | MR | Zbl

[24] V. Diekert and A. Muscholl, Solvability of equations in free partially commutative groups is decidable, in Automata, Languages and Programming, Lect. Notes Comput. Sci., vol. 2076, Springer, 2001, pp. 543-554. | MR | Zbl

[25] Druţu, C., Sapir, M. (2005) Tree-graded spaces and asymptotic cones of groups. Topology 44: pp. 959-1058 | MR | Zbl

[26] Druţu, C., Sapir, M. (2007) Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups. Adv. Math. 217: pp. 1313-1367 | MR | Zbl

[27] Dunwoody, M., Sageev, M. (1999) JSJ-splittings for finitely presented groups over slender groups. Invent. Math. 135: pp. 25-44 | MR | Zbl

[28] Epstein, D., Cannon, J., Holt, D., Levy, S., Paterson, M., Thurston, W. (1992) Word Processing in Groups. Jones and Bartlett, Boston | MR | Zbl

[29] Farb, B. (1998) Relatively hyperbolic groups. Geom. Funct. Anal. 8: pp. 810-840 | MR | Zbl

[30] Forester, M. (2003) On uniqueness of JSJ decompositions of finitely generated groups. Comment. Math. Helv. 78: pp. 740-751 | MR | Zbl

[31] Fujiwara, K., Papasoglu, P. (2006) JSJ decompositions of finitely presented groups and complexes of groups. Geom. Funct. Anal. 16: pp. 70-125 | MR | Zbl

[32] V. Gerasimov, Detecting connectedness of the boundary of a hyperbolic group, preprint.

[33] Grigorchuk, R., Lysenok, I. (1992) A description of solutions of quadratic equations in hyperbolic groups. Int. J. Algebra Comput. 2: pp. 237-274 | MR | Zbl

[34] M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75-263. | MR | Zbl

[35] Groves, D. (2005) Limits of certain CAT(0) groups, I: Compactification. Algebr. Geom. Topol. 5: pp. 1325-1364 | MR | Zbl

[36] D. Groves, Limit groups for relatively hyperbolic groups, I: The basic tools, preprint. | MR | Zbl

[37] Groves, D. (2005) Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams. Geom. Topol. 9: pp. 2319-2358 | MR | Zbl

[38] Grunewald, F., Segal, D. (1980) Some general algorithms. II. Nilpotent groups. Ann. Math. (2) 112: pp. 585-617 | MR | Zbl

[39] Guirardel, V. (2004) Limit groups and groups acting freely on R n -trees. Geom. Topol. 8: pp. 1427-1470 | MR | Zbl

[40] Hruska, G.C., Kleiner, B. (2005) Hadamard spaces with isolated flats, with an appendix by the authors and Mohamad Hindawi. Geom. Topol. 9: pp. 1501-1538 | MR | Zbl

[41] Hummel, C. (1998) Rank one lattices whose parabolic isometries have no rational part. Proc. Amer. Math. Soc. 126: pp. 2453-2458 | MR | Zbl

[42] Kharlampovich, O., Miasnikov, A. (2005) Effective JSJ decompositions. Groups, Languages and Algorithms. Amer. Math. Soc., Providence, RI, pp. 87-212 | MR | Zbl

[43] Kharlampovich, O., Miasnikov, A. (2006) Elementary theory of free non-abelian groups. J. Algebra 302: pp. 451-552 | MR | Zbl

[44] Levitt, G. (2005) Automorphisms of hyperbolic groups and graphs of groups. Geom. Dedicata 114: pp. 49-70 | MR | Zbl

[45] Levitt, G. (2005) Characterizing rigid simplicial actions on trees. Geometric Methods in Group Theory. Amer. Math. Soc., Providence, RI, pp. 27-33 | MR | Zbl

[46] Lyndon, R.C., Schupp, P.E. (1977) Combinatorial Group Theory. Springer, Berlin | MR | Zbl

[47] I. Lysenok, Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat., 53 (1989), 814-832, 912; transl. in Math. USSR-Izv., 35 (1990), 145-163. | MR | Zbl

[48] G. S. Makanin, Decidability of the universal and positive theories of a free group (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 735-749; transl. in Math. USSR-Izv., 25 (1985), 75-88. | MR | Zbl

[49] Miller Iii, C.F. (1971) On Group Theoretic Decision Problems and Their Classification. Princeton University Press, Princeton | MR | Zbl

[50] Miller Iii, C.F. (1992) Decision problems for groups - survey and reflections. Algorithms and Classification in Combinatorial Group Theory (Berkeley, CA, 1989). Springer, New York, pp. 1-59 | MR | Zbl

[51] Osin, D.V. (2006) Relatively Hyperbolic Groups: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems. Amer. Math. Soc., Providence, RI | MR | Zbl

[52] P. Papasoglu, An algorithm detecting hyperbolicity, in Geometric and Computational Perspectives on Infinite Groups (Minneapolis, MN and New Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 25, Amer. Math. Soc., Providence, RI, 1996, pp. 193-200. | MR | Zbl

[53] Paulin, F. (1988) Topologie de Gromov équivariante, structures hyperbolique et arbres réels. Invent. Math. 94: pp. 53-80 | MR | Zbl

[54] Rabin, M.O. (1958) Recursive unsolvability of group theoretic problems. Ann. Math. 67: pp. 172-194 | MR | Zbl

[55] D. Rebbechi, Algorithmic Properties of Relatively Hyperbolic Groups, PhD Thesis, ArXiv math.GR/0302245. | MR

[56] Rips, E., Sela, Z. (1995) Canonical representatives and equations in hyperbolic groups. Invent. Math. 120: pp. 489-512 | MR | Zbl

[57] Rips, E., Sela, Z. (1997) Cyclic splittings of finitely presented groups and the canonical JSJ decomposition. Ann. Math. (2) 146: pp. 53-104 | MR | Zbl

[58] V. Roman'Kov, Universal theory of nilpotent groups, Mat. Zametki, 25 (1979), 487-495, 635. | MR | Zbl

[59] Segal, D. (1990) Decidable properties of polycyclic groups. Proc. Lond. Math. Soc., III. Ser. 61: pp. 497-528 | MR | Zbl

[60] Sela, Z. (1995) The isomorphism problem for hyperbolic groups I. Ann. Math. (2) 141: pp. 217-283 | MR | Zbl

[61] Sela, Z. (1997) Acylindrical accessibility for groups. Invent. Math. 129: pp. 527-565 | MR | Zbl

[62] Sela, Z. (1997) Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II. Geom. Funct. Anal. 7: pp. 561-593 | MR | Zbl

[63] Sela, Z. (2003) Diophantine geometry over groups, I: Makanin-Razborov diagrams. Publ. Math., Inst. Hautes Étud. Sci. 93: pp. 31-105 | Numdam | MR | Zbl

[64] Sela, Z. (2006) Diophantine geometry over groups, VI: The elementary theory of a free group. Geom. Funct. Anal. 16: pp. 707-730 | MR | Zbl

[65] Z. Sela, Diophantine Geometry over Groups, VIII: Elementary Theory of Hyperbolic Groups, preprint (2002). | MR | Zbl

[66] Serre, J.-P. (1980) Trees. Springer, Berlin | MR