Preface -- Introduction -- Acknowledgments -- 1 A glance of the classical theory -- 2 Complex hyperbolic geometry -- 3 Complex Kleinian groups -- 4 Geometry and dynamics of automorphisms of P2C -- 5 Kleinian groups with a control group -- 6 The limit set in dimension two -- 7 On the dynamics of discrete subgroups of PU(n,1) -- 8 Projective orbifolds and dynamics in dimension two -- 9 Complex Schottky groups -- 10 Kleinian groups and twistor theory -- Bibliography -- Index.  .

This monograph lays down the foundations of the theory of complex Kleinian groups, a “newborn” area of mathematics whose origin can be traced 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 themselves be regarded as groups of holomorphic automorphisms of the complex projective line CP1. When we go 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 differ 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, about an area of mathematics that is still in its infancy, and this is the focus of study in this monograph. It brings together several important areas of mathematics, e.g. classical Kleinian group actions, complex hyperbolic geometry, crystallographic groups and the uniformization problem for complex manifolds.