Geometrical Finiteness for Hyperbolic Groups

@article{Bowditch1993GeometricalFF,
  title={Geometrical Finiteness for Hyperbolic Groups},
  author={Brian H. Bowditch},
  journal={Journal of Functional Analysis},
  year={1993},
  volume={113},
  pages={245-317}
}
  • B. Bowditch
  • Published 1 May 1993
  • Mathematics
  • Journal of Functional Analysis
Abstract In this paper, we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension. We prove the equivalence of various definitions of geometrical finiteness, and describe the geometry of fundamental domains. We give a complete account of when Dirichiet domains are finite-sided. 

Ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3-manifolds

<abstract abstract-type="TeX"><p>We show that geometrically finite Kleinian groups are dense in the boundary of the quasiconformal deformation space of any geometrically finite Kleinian group.

Spaces of geometrically finite representations

We explore conditions under which the property of geometrical finiteness is open among type-preserving representations of a given group into the group of isometries of hyperbolic n-space. We give

On the Patterson–Sullivan Measure for Geometrically Finite Groups Acting on Complex or Quaternionic Hyperbolic Space

The goal of this paper is to provide a tool, the Global Measure Formula, that will facilitate the study of the limit set of discrete geometrically finite groups of isometries of the rank one

An infinitely generated intersection of geometrically finite hyperbolic groups

Two discrete, geometrically finite subgroups of the isometrics of hyperbolic n-space (n > 4) are defined whose intersection is infinitely generated. This settles, in dimensions 4 and above, a

Dirichlet polyhedra for cyclic groups in complex hyperbolic space

We prove that the Dirichlet fundamental polyhedron for a cyclic group generated by a unipotent or hyperbolic element γ acting on complex hyperbolic n-space centered at an arbitrary point w is bounded

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

For geometrically finite hyperbolic manifolds Γ\ℍn+1, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole

Hausdorff Dimension and Limits of Kleinian Groups

Abstract. In this paper we prove that if M is a compact, hyperbolizable 3-manifold, which is not a handlebody, then the Hausdorff dimension of the limit set is continuous in the strong topology on

Rigidity of limit sets for nonplanar geometrically finite Kleinian groups of the second kind

We consider the relation between geometrically finite groups and their limit sets in infinite-dimensional hyperbolic space. Specifically, we show that a rigidity theorem of Susskind and Swarup (Am J

On limits of tame hyperbolic 3-manifolds

We show that if a purely hyperbolic Kleinian group is the strong limit of a sequence of topologically tame purely hyperbolic Kleinian groups, then it is topologically tame. We then apply this result
...