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Birkhäuser

Complex Kleinian Groups

  • Book
  • © 2013

Overview

Authors:
  1. Angel Cano

    1. , Instituto de Matemáticas, UNAM, Unidad Cuernavaca, Cuernavaca, Mexico

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  2. Juan Pablo Navarrete

    1. , Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mérida, Mexico

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    You can also search for this author in PubMed   Google Scholar

  3. José Seade

    1. , Facultad de Matemáticas, UNAM, Unidad Cuernavaca, Cuernavaca, Mexico

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    You can also search for this author in PubMed   Google Scholar

  • 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

Part of the book series: Progress in Mathematics (PM, volume 303)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xx
    Download chapter PDF

  2. A Glance at the Classical Theory

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 1-40

  3. Complex Hyperbolic Geometry

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 41-76

  4. Complex Kleinian Groups

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 77-92

  5. Geometry and Dynamics of Automorphisms of \(\mathbb{P}^{2}_\mathbb{C}\)

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 93-118

  6. Kleinian Groups with a Control Group

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 119-136

  7. The Limit Set in Dimension 2

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 137-143

  8. On the Dynamics of Discrete Subgroups of PU(n, 1)

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 145-166

  9. Projective Orbifolds and Dynamics in Dimension 2

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 167-194

  10. Complex Schottky Groups

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 195-229

  11. Kleinian Groups and Twistor Theory

    • Angel Cano, Juan Pablo Navarrete, José Seade
    Pages 231-251

  12. Back Matter

    Pages 253-271
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Keywords

About this book

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.​

Reviews

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)

Authors and Affiliations

  • , Instituto de Matemáticas, UNAM, Unidad Cuernavaca, Cuernavaca, Mexico

    Angel Cano

  • , Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mérida, Mexico

    Juan Pablo Navarrete

  • , Facultad de Matemáticas, UNAM, Unidad Cuernavaca, Cuernavaca, Mexico

    José Seade

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