Complex Kleinian Groups
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Beschreibung
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.
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.
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
Autor*in
Angel Cano
Themen in »Complex Kleinian Groups«
Kleinian groups complex hyperbolic geometry discontinuity region equicontinuity limit set
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From the reviews:
“The book is written in a clear, accessible manner and selected chapters could easily serve as a text for a graduate course on this topic. It also brings together many results published by the authors, their collaborators and others on this topic, as well as giving open questions and directions for future research.” (John R. Parker, Mathematical Reviews, February, 2014)
“A wonderful monograph on complex Kleinian groups which is of great interest for researchers and graduate students in the area of complex Kleinian groups and hyperbolic geometry. Each individual chapter is a unit by itself. … The monograph is very well written and structured. … I strongly recommend it.” (Gerhard Rosenberger, zbMATH, Vol. 1267, 2013)()