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EBOOK
Author Cano, Angel.
Title Complex Kleinian groups / Angel Cano, Juan Pablo Navarrete, Jose Seade.
Imprint Basel ; New York : Birkhauser : Springer, 2013.

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Series Progress in mathematics ; 303
Progress in mathematics (Boston, Mass.) ; v. 303.
Subject Kleinian groups.
Alt Name Navarrete, Juan Pablo.
Seade, J. (José)
Description 1 online resource (xx, 271 pages) : illustrations.
Bibliography Note Includes bibliographical references (pages 253-267) and index.
Contents A Glance at the Classical Theory -- Complex Hyperbolic Geometry -- Complex Kleinian Groups -- Geometry and Dynamics of Automorphisms of P₂C -- Kleinian Groups with a Control Group -- The Limit Set in Dimension 2 -- On the Dynamics of Discrete Subgroups of PU(n, 1) -- Projective Orbifolds and Dynamics in Dimension 2 -- Complex Schottky Groups -- Kleinian Groups and Twistor Theory.
Machine generated contents note: 1. Glance at the Classical Theory -- 1.1. Isometries of hyperbolic n-space. The conformal group -- 1.1.1. Poincare models for hyperbolic space -- 1.1.2. Mobius groups in dimensions 2 and 3 -- 1.1.3. Geometric classification of the elements in Iso+ (HnR) -- 1.1.4. Isometric spheres -- 1.2. Discrete subgroups -- 1.2.1. Properly Discontinuous Actions -- 1.2.2. limit set and the discontinuity region -- 1.2.3. Fundamental domains -- 1.2.4. Fuchsian groups -- 1.2.5. Kleinian and Schottky groups -- 1.3. Rigidity and ergodicity -- 1.3.1. Moore's Ergodicity Theorem -- 1.3.2. Mostow's rigidity theorem -- 1.3.3. On the Patterson-Sullivan measure -- 1.3.4. Sullivan's theorem on nonexistence of invariant lines -- 2. Complex Hyperbolic Geometry -- 2.1. Some basic facts on Projective geometry -- 2.2. Complex hyperbolic geometry. The ball model -- 2.2.1. Totally geodesic subspaces -- 2.2.2. Bisectors and spines -- 2.3. Siegel domain model -- 2.3.1. Heisenberg geometry and horospherical coordinates -- 2.3.2. geometry at infinity -- 2.4. Isometries of the complex hyperbolic space -- 2.4.1. Complex reflections -- 2.4.2. Dynamical classification of the elements in PU (2, 1) -- 2.4.3. Traces and conjugacy classes in SU (2, 1) -- 2.5. Complex hyperbolic Kleinian groups -- 2.5.1. Constructions of complex hyperbolic lattices -- 2.5.2. Other constructions of complex hyperbolic Kleinian groups -- 2.6. Chen-Greenberg limit set -- 3. Complex Kleinian Groups -- 3.1. limit set: an example -- 3.2. Complex Kleinian groups: definition and examples -- 3.3. Limit sets: definitions and some basic properties -- 3.3.1. Limit set of Kulkarni -- 3.3.2. Elementary groups -- 3.4. On the subgroups of the affine group -- 3.4.1. Fundamental groups of Hopf manifolds -- 3.4.2. Fundamental groups of complex tori -- 3.4.3. suspension or cone construction -- 3.4.4. Example of elliptic affine surfaces -- 3.4.5. Fundamental groups of Inoue surfaces -- 3.4.6. group induced by a hyperbolic toral automorphism -- 3.4.7. Crystallographic and complex affine reflection groups -- 4. Geometry and Dynamics of Automorphisms of P2C -- 4.1. qualitative view of the classification problem -- 4.2. Classification of the elements in PSL (3, C) -- 4.2.1. Elliptic Transformations in PSL (3, C) -- 4.2.2. Parabolic Transformations in PSL (3, C) -- 4.2.3. Loxodromic Transformations in PSL (3, C) -- 4.3. classification theorems -- 5. Kleinian Groups with a Control Group -- 5.1. PSL (2, C) revisited: nondiscrete subgroups -- 5.1.1. Main theorems for nondiscrete subgroups of PSL (2, C) -- 5.2. Some basic examples -- 5.3. Elementary groups -- 5.3.1. Groups whose equicontinuity region is the whole sphere -- 5.3.2. Classification of elementary groups -- 5.4. Consequences of the classification theorem -- 5.5. Controllable and control groups: definitions -- 5.5.1. Suspensions -- 5.6. Controllable groups -- 5.7. Groups with control -- 5.8. On the limit set -- 5.8.1. limit set for suspensions extended by a group -- 5.8.2. discontinuity region for some weakly semi-controllable groups -- 6. Limit Set in Dimension 2 -- 6.1. Montel's theorem in higher dimensions -- 6.2. Lines and the limit set -- 6.3. limit set is a union of complex projective lines -- 7. On the Dynamics of Discrete Subgroups of PU (n, 1) -- 7.1. Discrete subgroups of PU (n, 1) revisited -- 7.2. Some properties of the limit set -- 7.3. Comparing the limit sets ΛKul (G) and ΛCG (G) -- 7.4. Pseudo-projective maps and equicontinuity -- 7.5. On the equicontinuity region -- 7.6. Geometric Applications -- 7.6.1. Kobayashi Metric -- 7.6.2. Complex Hyperbolic groups and k-chains -- 7.7. two-dimensional case revisited -- 8. Projective Orbifolds and Dynamics in Dimension 2 -- 8.1. Geometric structures and the developing map -- 8.1.1. Projective structures on manifolds -- 8.1.2. developing map and holonomy -- 8.2. Real Projective Structures and Discrete Groups -- 8.2.1. Projective structures on real surfaces -- 8.2.2. On divisible convex sets in real projective space -- 8.3. Projective structures on complex surfaces -- 8.4. Orbifolds -- 8.4.1. Basic notions on orbifolds -- 8.4.2. Description of the compact (P2C, PSL3 (C))-orbifolds -- 8.5. Discrete groups and divisible sets in dimension 2 -- 8.6. Elementary quasi-cocompact groups of PSL (3, C) -- 8.7. Nonelementary affine groups -- 8.8. Concluding remarks -- 8.8.1. Summary of results for quasi-cocompact groups -- 8.8.2. Comments and open questions -- 9. Complex Schottky Groups -- 9.1. Examples of Schottky groups -- 9.1.1. Seade-Verjovsky complex Schottky groups -- 9.1.2. Nori's construction of complex Schottky groups -- 9.2. Schottky groups: definition and basic facts -- 9.3. On the limit set and the discontinuity region -- 9.3.1. Quotient spaces of the region of discontinuity -- 9.3.2. Hausdorff dimension and moduli spaces -- 9.4. Schottky groups do not exist in even dimensions -- 9.4.1. On the dynamics of projective transformations -- 9.4.2. Nonrealizability of Schottky groups in PSL (2n + 1, C) -- 9.5. Complex kissing-Schottky groups -- 9.6. "zoo" in dimension 2 -- 9.7. Remarks on the uniformisation of projective 3-folds -- 10. Kleinian Groups and Twistor Theory -- 10.1. twistor fibration -- 10.1.1. twistor fibration in dimension 4 -- 10.1.2. twistor fibration in higher dimensions -- 10.2. Canonical Lifting -- 10.2.1. Lifting Conf+ (S4) to PSL (4, C) -- 10.2.2. Canonical lifting in higher dimensions -- 10.3. Complex Kleinian groups on P3C -- 10.4. Kleinian groups and twistor spaces in higher dimensions -- 10.5. Patterson-Sullivan measures on twistor spaces -- 10.6. Some remarks.
Summary 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, Poincare, 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 comp.
Note Print version record.
ISBN 9783034804813 (ebk.)
3034804814 (ebk.)
9781283909686
1283909685
9783034804806
3034804806
OCLC # 823386546
Additional Format Print version: Cano, Angel. Complex Kleinian groups. Basel ; New York : Birkhauser : Springer, 2013 9783034804806 (DLC) 2012951146 (OCoLC)793570662