Abstract

A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type-preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current state of the art of the theory of complex hyperbolic quasi-Fuchsian groups.

Introduction

The purpose of this paper is to outline what is known about the complex hyperbolic analogue of quasi-Fuchsian groups. Discrete groups of complex hyperbolic isometries have not been studied as widely as their real hyperbolic counterparts. Nevertheless, they are interesting to study and should be more widely known. The classical theory of quasi-Fuchsian groups serves as a model for the complex hyperbolic theory, but results do not usually generalise in a straightforward way. This is part of the interest of the subject.

Complex hyperbolic Kleinian groups were first studied by Picard at about the same time as Poincaré was developing the theory of Fuchsian and Kleinian groups. In spite of work by several other people, including Giraud and Cartan, the complex hyperbolic theory did not develop as rapidly as the real hyperbolic theory. So, by the time Ahlfors and Bers were laying the foundations for the theory of quasi-Fuchsian groups, complex hyperbolic geometry was hardly studied at all. Later, work of Chen and Greenberg and of Mostow on symmetric spaces led to a resurgence of interest in complex hyperbolic discrete groups.