  We study geometry, topology and deformation spaces of noncompact complex
hyperbolic manifolds (geometrically finite, with variable negative curvature),
whose properties make them surprisingly different from real hyperbolic
manifolds with constant negative curvature. This study uses an interaction
between K\"ahler geometry of the complex hyperbolic space and the contact
structure at its infinity (the one-point compactification of the Heisenberg
group), in particular an established structural theorem for discrete group
actions on nilpotent Lie groups.
