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A topological characterisation of relatively hyperbolic groups. (English) Zbl 1043.20020
The notion of a relatively hyperbolic group was introduced by M. Gromov and has been elaborated on in various papers. As hyperbolic groups generalize fundamental groups of compact hyperbolic manifolds, the relatively hyperbolic groups generalize fundamental groups of hyperbolic manifolds of finite volume. The paper gives dynamical and topological properties characterising relatively hyperbolic groups in terms of their boundaries.
The following theorem is proved: Suppose that $$M$$ is a non-empty, perfect and metrisable compactum, and $$\Gamma$$ is a convergence group acting on $$M$$ such that $$M$$ consists only of conical limit points and bounded parabolic points. Suppose also that the quotient of the set of bounded parabolic points by $$\Gamma$$ is finite and the stabiliser of each bounded parabolic point is finitely generated. Then $$\Gamma$$ is hyperbolic relative to the set of its maximal parabolic subgroups and $$M$$ is equivariantly homeomorphic to the boundary of $$\Gamma$$.
The main idea of the proof of the above theorem is the following. The author constructs a “system of annuli”, generalizing one given in a paper of B. H. Bowditch [J. Am. Math. Soc. 11, No. 3, 643-667 (1998; Zbl 0906.20022)], which gives rise to a path hyperbolic quasimetric on the set of distinct triples union of the set of bounded parabolic points. Next, using the geometrically finite action the author constructs a graph satisfying all properties of relative hyperbolicity.

MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory 57M07 Topological methods in group theory 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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References:
 [1] Proc. Nat. Acad. Sci. U.S.A. 55 pp 251– (1966) [2] [Bea] A. F. Beardon, The geometry of discrete groups, Grad. Texts Math. 91, Springer-Verlag, Berlin-New York, 1983. [3] Bea A. F., Acta Math. 132 pp 1– (1974) [4] Ber L, Ann. Math. 91 pp 570– (1970) [5] [Bo1] B. H. Bowditch, Notes on Gromov’shyperbolicity criterion for path-metric spaces, in: Group theory from a geometrical viewpoint, E. Ghys, A. Haefliger, A. Verjovsky, eds., World Scientific (1991), 64-167. [6] Bo B. H, J. Funct. Anal. 113 pp 245– (1993) [7] Bo B. H, Duke Math. J. 77 pp 229– (1995) [8] Bo B. H, J. Amer. Math. Soc. 11 pp 643– (1998) [9] [Bo5] B. H. Bowditch, Convergence groups and configuration spaces, in: Group theory down under, J. Cossey, C. F. Miller, W. D. Neumann, M. Shapiro, eds., de Gruyter (1999), 23-54. [10] [Bo6] B. H. Bowditch, Relatively hyperbolic groups, preprint, Southampton 1997. [11] [Bo7] B. H. Bowditch, Groups acting on Cantor sets and the end structure of graphs, preprint, Southampton 2000. [12] Fa B., Geom. Funct. Anal. 8 pp 810– (1998) [13] Fr E. M, Conform. Geom. Dynam. 1 pp 13– (1997) [14] Ge F. W., Proc. London Math. Soc. 55 pp 331– (1987) [15] [GhH] E. Ghys, P. de la Harpe, eds., Sur les groupes hyperboliques d’apres Mikhael Gromov, Progr. Math. 83, Birkh user, 1990. [16] Gre L., Ann. Math. 84 pp 433– (1966) [17] [Gro] M. Gromov, Hyperbolic groups, in: Essays in Group Theory, S. M. Gersten, ed., M.S.R.I. Publ. 8, Springer-Verlag (1987), 75-263. [18] Duke Math. J. 2 pp 530– (1936) [19] Mar A, Ann. Math. J. 99 pp 383– (1974) [20] [MatT] K. Matsuzaki, M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Sci. Publ.1998. · Zbl 0892.30035 [21] Nicholls, Proc. London Math. Soc. pp 143– (1989) [22] Otal, Rev. Math. Iberoamer. 8 pp 441– (1992) [23] J. London Math. Soc. 54 pp 50– (1996) [24] Su D., Acta. Math. 153 pp 259– (1984) [25] Sz A., Michigan Math. J. 45 pp 611– (1998) [26] [Th] W. P. Thurston, The Geometry and Topology of 3-Manifolds, notes, Princeton Univ. Maths. Department, 1979. [27] Tu P., New Zealand J. Math. 23 pp 157– (1994) [28] Tu P, Math. 501 pp 71– (1998)
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