Geometrically Finite Complex Hyperbolic Manifolds

Abstract

The aim of this paper is to study geometry and topology of geometrically finite complex hyperbolic manifolds, especially their ends, as well as geometry of their holonomy groups. This study is based on our structural theorem for discrete groups acting on Heisenberg groups, on the fiber bundle structure of Heisenberg manifolds, and on the existence of finite coverings of a geometrically finite manifold such that their parabolic ends have either Abelian or 2-step nilpotent holonomy. We also study an interplay between Kähler geometry of complex hyperbolic n-manifolds and Cauchy–Riemannian geometry of their boundary (2n-1)-manifolds at infinity, and this study is based on homotopy equivalence of manifolds and isomorphism of fundamental groups.