Relatively hyperbolic Groups

In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

[1]  A. Haefliger,et al.  Group theory from a geometrical viewpoint , 1991 .

[2]  A. Yaman,et al.  A topological characterisation of relatively hyperbolic groups , 2004 .

[3]  Inna Bumagin.  On definitions of relatively hyperbolic groups , 2004 .

[4]  L V Ahlfors.  Fundamental polyhedrons and limit point sets of kleinian groups. , 1966, Proceedings of the National Academy of Sciences of the United States of America.

[5]  François Dahmani.  Les groupes relativement hyperboliques et leurs bords , 2003 .

[6]  Brian H. Bowditch,et al.  Geometrical Finiteness for Hyperbolic Groups , 1993 .

[7]  Benson Farb,et al.  Relatively Hyperbolic Groups , 1998 .

[8]  É. Ghys,et al.  Sur Les Groupes Hyperboliques D'Apres Mikhael Gromov , 1990 .

[9]  G. A. Hedlund,et al.  Fuchsian groups and transitive horocycles , 1936 .

[10]  Brian H. Bowditch,et al.  A topological characterisation of hyperbolic groups , 1998 .

[11]  Albert Marden,et al.  The Geometry of Finitely Generated Kleinian Groups , 1974 .

[12]  L. Greenberg,et al.  Fundamental Polyhedra for Kleinian Groups , 1966 .

[13]  Daniel Groves,et al.  Dehn filling in relatively hyperbolic groups , 2006 .

[14]  Cornelia Drutu,et al.  Tree-graded spaces and asymptotic cones of groups , 2004 .

[15]  Brian H. Bowditch,et al.  Geometrical finiteness with variable negative curvature , 1995 .

[16]  Yair N. Minsky,et al.  Geometry of the complex of curves I: Hyperbolicity , 1998, math/9804098.

[17]  Victor Gerasimov.  Expansive Convergence Groups are Relatively Hyperbolic , 2009 .

[18]  A. I. MathematicaVolumen.  Negatively Curved Groups Have the Convergence Property I , 1995 .

[19]  Andrzej Szczepański,et al.  Relatively hyperbolic groups. , 1998 .

[20]  Brian H. Bowditch,et al.  Discrete parabolic groups , 1993 .

[21]  Alan F. Beardon,et al.  Limit points of Kleinian groups and finite sided fundamental polyhedra , 1974 .

[22]  Brian H. Bowditch.  Boundaries of geometrically finite groups , 1999 .

[23]  Pekka Tukia,et al.  On isomorphisms of geometrically finite Möbius groups , 1985 .

[24]  Pekka Tukia,et al.  Conical limit points and uniform convergence groups , 1998 .

[25]  M. J. Dunwoody.  The accessibility of finitely presented groups , 1985 .

[26]  S. M. Gersten,et al.  Subgroups of Word Hyperbolic Groups in Dimension 2 , 1996 .

[27]  Athanase Papadopoulos,et al.  Géométrie et théorie des groupes , 1990 .

[28]  Eric Freden,et al.  Properties of convergence groups and spaces , 1997 .

[29]  Brian H. Bowditch.  A class of incomplete non-positively curved manifolds , 1996 .

[30]  Max L. Warshauer,et al.  Lecture Notes in Mathematics , 2001 .

[31]  D. V. Osin.  Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems , 2004 .

[32]  Brian H. Bowditch.  Peripheral splittings of groups , 2001 .