This monograph lays down the foundations of the theory of complex
Kleinian groups, a newly born area of mathematics whose origin traces
back to the work of Riemann, Poincaré, Picard and many others. Kleinian
groups are, classically, discrete groups of conformal automorphisms of
the Riemann sphere, and these can be regarded too as being groups of
holomorphic automorphisms of the complex projective line CP1.
When going into higher dimensions, there is a dichotomy: Should we look
at conformal automorphisms of the n-sphere?, or should we look at
holomorphic automorphisms of higher dimensional complex projective
spaces? These two theories are different in higher dimensions. In the
first case we are talking about groups of isometries of real hyperbolic
spaces, an area of mathematics with a long-standing tradition. In the
second case we are talking about an area of mathematics that still is in
its childhood, and this is the focus of study in this monograph. This
brings together several important areas of mathematics, as for instance
classical Kleinian group actions, complex hyperbolic geometry,
chrystallographic groups and the uniformization problem for complex
manifolds.