Progress in Mathematics

Complex Kleinian Groups

Authors: Cano, Angel, Navarrete, Juan Pablo, José Antonio, Seade Kuri

  • 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
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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)

Table of contents (10 chapters)

  • A Glance at the Classical Theory

    Cano, Angel (et al.)

    Pages 1-40

  • Complex Hyperbolic Geometry

    Cano, Angel (et al.)

    Pages 41-76

  • Complex Kleinian Groups

    Cano, Angel (et al.)

    Pages 77-92

  • Geometry and Dynamics of Automorphisms of $$\mathbb{P}^{2}_\mathbb{C}$$

    Cano, Angel (et al.)

    Pages 93-118

  • Kleinian Groups with a Control Group

    Cano, Angel (et al.)

    Pages 119-136

Buy this book

eBook $79.99
price for USA (gross)
  • ISBN 978-3-0348-0481-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $109.00
price for USA
  • ISBN 978-3-0348-0480-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $109.00
price for USA
  • ISBN 978-3-0348-0805-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Complex Kleinian Groups
Authors
Series Title
Progress in Mathematics
Series Volume
303
Copyright
2013
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel
eBook ISBN
978-3-0348-0481-3
DOI
10.1007/978-3-0348-0481-3
Hardcover ISBN
978-3-0348-0480-6
Softcover ISBN
978-3-0348-0805-7
Series ISSN
0743-1643
Edition Number
1
Number of Pages
XX, 272
Topics