# 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} }

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.

## 226 Citations

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

- Mathematics
- 2004

<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

- Mathematics
- 1995

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

- Mathematics
- 2003

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

- Mathematics
- 2001

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

- Mathematics
- 1992

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

- Mathematics
- 2010

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…

The integral K-theoretic Novikov conjecture for geometrically finite groups

- Mathematics
- 2005

In this note, we prove the integral K-theoretic Novikov conjecture for geometrically finite discrete subgroups of semisimple Lie groups of rank 1, in particular geometrically finite Kleinian groups.

Hausdorff Dimension and Limits of Kleinian Groups

- Mathematics
- 1999

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…

Representations of polygons of finite groups

- Mathematics
- 2005

We construct discrete and faithful representations into the isometry group of a hyperbolic space of the fundamental groups of acute negatively curved evensided polygons of finite groups.