Ebook: Complex Kleinian Groups

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.​

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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, Poincar?, 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.

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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, Poincar?, 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.

Content:
Front Matter....Pages i-xx
A Glance at the Classical Theory....Pages 1-40
Complex Hyperbolic Geometry....Pages 41-76
Complex Kleinian Groups....Pages 77-92
Geometry and Dynamics of Automorphisms of $mathbb{P}^{2}_mathbb{C}$ ....Pages 93-118
Kleinian Groups with a Control Group....Pages 119-136
The Limit Set in Dimension 2....Pages 137-143
On the Dynamics of Discrete Subgroups of PU(n, 1)....Pages 145-166
Projective Orbifolds and Dynamics in Dimension 2....Pages 167-194
Complex Schottky Groups....Pages 195-229
Kleinian Groups and Twistor Theory....Pages 231-251
Back Matter....Pages 253-271

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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, Poincar?, 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.

Content:
Front Matter....Pages i-xx
A Glance at the Classical Theory....Pages 1-40
Complex Hyperbolic Geometry....Pages 41-76
Complex Kleinian Groups....Pages 77-92
Geometry and Dynamics of Automorphisms of $mathbb{P}^{2}_mathbb{C}$ ....Pages 93-118
Kleinian Groups with a Control Group....Pages 119-136
The Limit Set in Dimension 2....Pages 137-143
On the Dynamics of Discrete Subgroups of PU(n, 1)....Pages 145-166
Projective Orbifolds and Dynamics in Dimension 2....Pages 167-194
Complex Schottky Groups....Pages 195-229
Kleinian Groups and Twistor Theory....Pages 231-251
Back Matter....Pages 253-271
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