Discrete Groups in Space and Uniformization Problems

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Springer Science & Business Media, Jun 30, 1991 - Mathematics - 482 pages
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A revised and substantially enlarged edition of the Russian book Discrete transformation groups and manifold structures published by Nauka in 1983, this volume presents a comprehensive treatment of the geometric theory of discrete groups and the associated tessellations of the underlying space. Also
 

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Contents

Conformal mappings
1
2 Quasiconformal mappings and isometry
2
3 Möbius transformations
4
4 Automorphisms of a ball and halfspace
11
5 Fixed points of Möbius transformations
15
6 Invariant domains for Möbius subgroups
16
Geometrical structures on manifolds
19
2 Homology and MayerVletoris sequences
22
5 Invariant components for geometrically finite Kleinian groups
154
6 The geometry of the Nielsen hull boundary
168
Geometric and algebraic structures of discrete groups
191
2 Free amalgamated products of Kleinian groups
204
3 HNNextensions of Kleinian groups Examples
212
4 Cayley graph and combinatorial structure of a discrete group
223
5 Geometric and algebraic convergence of groups
236
Sullivans ergodic theorems
241

3 Spaces of constant curvature
26
4 Representation of the Möbius group in the matrix group
32
5 𝒱structures in manifolds
34
6 Developing map and holonomy
41
7 Orbifords and their 𝒱structures
43
General properties of discontinuous groups
51
2 Action of a group on the discontinuity set
54
3 Some classes of Kleinian groups
61
4 Fundamental domains
64
the nonmeasurability effect and density property
73
Discrete groups of hyperbolic Isometries
77
2 Arithmetic construction of cocompact hyperbolic groups
85
3 Universal bounds of hyperbolic manifold sizes
87
4 Margulis lemma and hyperbolic manifold decomposition
93
5 Precisely invariant horoballs and parabolic 𝜴peaks
101
6 Crystallographic reflection groups and arithmetic
108
Geometrically finite discrete Möbius groups
117
2 The convex Nielsen hull of the limit set
122
3 The limit set of a geometrically finite group
126
4 Finiteness theorems for discrete Möbius groups
141
Kleinian manifolds
251
2 Topological aspects of combination theorems
259
3 Kleinian manifold of a function group
263
4 Isomorphism theorems
283
5 Geometrically tame ends Kleinian manifold compactification and the Ahlfors problem on limitset measure
290
Uniformization of manifolds
307
2 Gromovs and Thurstons geometric uniformization concepts
310
3 Thurstons hyperbolization theorem and the volumes of hyperbolic manifolds
318
4 Complexity of 3manifolds integrable Hamiltonian systems and hyperbolic volumes
328
5 Conformal uniformization of manifolds
333
6 Conformal uniformization of doubles of atoroidal manifolds
344
Deformation spaces of groups and structures
355
2 Rigidity theorems for complete hyperbolic structures and isometry groups
365
3 Deformation space for a Fuchsian group
381
4 The boundary of deformation space for a Fuchsian group
409
5 Deformations of Kleinian groups and conformal structures
415
References
425
Index
467
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