Topology of Kleinian Groups in Several Dimensions

1. Introduction The goal of this survey is to give an overview (from a topologist's prospective) of the theory of Kleinian groups in higher dimensions. The survey grew out of a series of lectures which I have given in the University of Maryland in Fall of 1991 and which I have decided to update and publish now, in the proceedings of the 3-rd Ahlfors-Bers Colloquium. This survey was greatly innuenced by the work of Lars Ahlfors, even the title was partly borrowed from his lecture notes 1]. There is a vast number of Kleinian groups in higher dimensions: there is no hope for a comprehensive structure theory similar to the theory of Kleinian subgroups of PSL(2; C). In this paper I organize higher-dimensional Kleinian groups according to the topology of their limit sets, trying to contrast and compare them with the Kleinian subgroups of PSL(2; C). In this setting, one of the key questions that I was trying to address is the interaction between the geometry and topology of the limit set and the algebraic and topological properties of the Kleinian group. The groups with zero-dimensional limit sets are relatively easy to understand (section 3). In the case of convex-cocompact groups with 1-dimensional limit sets, at least the topology of the limit sets is understood (see section 4), although their group-theoretic structure is a mystery. We know very little about Kleinian groups with higher-dimensional limit sets, thus I have restricted the discussion to Kleinian groups whose limit sets are topological spheres (section 5). I then discuss Ahlfors niteness theorem and its failure in higher dimensions (section 6). Next I consider the representation varieties of Kleinian groups and topological and geometrical constrains on Kleinian groups in higher dimensions. Lastly, I discuss generalizations of Kleinian groups: uniformly quasiconformal groups, fundamental groups of negatively curved manifolds and the convergence groups.