Conformal Geometry of Discrete Groups and Manifolds (De Gruyter Expositions in Mathematics Book 32)

"This is a useful and challenging book. The first three chapters may serve as a general introduction to the subject, the whole book as a source of information and as a reference for the area of conformal and hyperbolic geometry of manifolds and of Kleinian groups." __Mathematical Reviews__ Preface 1. Geometric Structures 1.1. (X, G)-structures on manifolds 1.2. Conformal geometry on the sphere 1.3. The hyperbolic space Hn 1.4. Lie subgroups of the Möbius group 1.5. Structure developments and holonomy homomorphisms 1.6. The eight 3-dimensional geometries 1.7. Four-dimensional geometries 1.8. Geometry of orbifolds Notes 2. Discontinuous Groups of Homeomorphisms 2.1. Convergence groups 2.2. Group action on the discontinuity set 2.3. Fundamental domains 2.4. Convex polyhedra and reflection groups 2.5. Discrete group action on the limit set Notes 3. Basics of Hyperbolic Groups and Manifolds 3.1. Margulis's Lemma and splittings of hyperbolic manifolds 3.2. Injectivity radius of hyperbolic manifolds 3.3. Thin cusp submanifolds 3.4. Precisely invariant horoballs 3.5. Group action on the set of horoballs 3.6. Convex hull constructions 3.7. Tessellations of manifolds by ideal hyperbolic polyhedra 3.8. Hyperbolic arithmetics 3.9. Arithmetic groups generated by reflections 3.10. Non-arithmetic groups of Gromov and Piatetski-Shapiro 3.11. Fibonacci manifolds Notes 4. Geometrical Finiteness 4.1. Classical finiteness for planar Kleinian groups 4.2. Geometrical finiteness in higher dimensions 4.3. Equivalent definitions of geometrical finiteness 4.4. Geometrically finite ends and coverings 4.5. Geometry of tessellations 4.6. Cayley graphs and geometric isomorphisms of discrete groups 4.7. Geometrical finiteness for discontinuity set components Notes 5. Kleinian Manifolds 5.1. Basic topology related to Kleinian manifolds 5.2. Topological aspects of combination theorems 5.3. Universal groups and Poincaré Conjecture 5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture 5.5. Kleinian n-manifolds and hyperbolic cobordisms 5.6. Finiteness problems for Kleinian n-manifolds Notes 6. Uniformization 6.1. Classical uniformization 6.2. Modern concepts of uniformization 6.3. Hyperbolization of manifolds and hyperbolic volumes 6.4. Uniformizable conformai structures 6.5. Conformai uniformization of "flat" connected sums 6.6. Conformai uniformization of Seifert manifolds 6.7. Torus sums of conformai structures 6.8. Canonical Riemannian metric on conformai manifolds Notes 7. Theory of Deformations 7.1. Deformations of geometric structures 7.2. Rigidity of hyperbolic structures 7.3. Quasi-Fuchsian structures: bendings 7.4. Quasi-Fuchsian structures: cone deformations 7.5. Bendings along surfaces with boundaries 7.6. Global properties of deformation spaces Notes Bibliography Index

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic.

Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urba?ski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

In addition to purely topological methods in the study of manifolds, last two decades results and especially Thurston's work [1-8] have shown that geometry also plays an important role in low-dimensional topology. This text presents a systematic account of conformal geometry of n-manifolds, as well as its Reimannian counterparts.