This book explores complex Kleinian groups, a new area of mathematics born in the work of Riemann, Poincaré, Picard and others. The author brings together classical Kleinian group actions, complex hyperbolic geometry, chrystallographic groups and more.

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 CP 1 . 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.

Lays down the foundations of a new field of mathematics including areas as important as real and complex hyperbolic geometry, discrete group actions in complex geometry and the uniformization problem First book of its kind in the literature Accessible to a wide audience Serves also as an introduction to the study of real and complex hyperbolic geometry Includes supplementary material: sn.pub/extras

Inhalt
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.