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  李群与李代数II...
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The first part of this book on Discrete Subgroups of Lie Groups is written by E.B. Vinberg. V.V. Gorbatsevich, and O.V. Shvartsman. Various types of discrete subgroups of Lie groups arise in the theory of functions of complex variables. arithmetic. geometry. and crystallography. S？nce the foundation of their general theory in the 50-60s of this century， considerable and in many respects exhaustive results were obtained. This development is reflected in this survey. Both semisimple and general Lie groups are considered. Partll on Cohomologies of Lie Groups and Lie Algebras is written by B.L. Feigin and D.B. Fuchs. It contains different definitions of cohomologies of Lie groups and (both finite-dimensional and some infinitedimensional) Lie algebras， the main methods of their calculation, and the results of these calculations.The book can be useful as a reference and research guide to graduate students and researchers in different areas of mathematics and theoretical physics.

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• Contents
  I. Discrete Subgroups of Lie Groups E. B. Vinberg, Y. Y. Gorbatsevich and O.Y. Shvartsman 1
  II. Cohomologies of Lie Groups and Lie Algebras B. L. Feigin and D. B. Fuchs 125
  Author Index 217
  Subject Index 221
  I. Discrete Subgroups of Lie Groups E. B. Vinberg , V V Gorbatsevich and O.V Shvartsman
  Contents
  Chapter 0. Introduction 5
  Chapter 1. Discrete Subgroups of Locally Compact Topological Groups 7
  1. The Simplest Properties of Lattices 7
  1.1. Definition of a Discrete Subgroup. Examples 7
  1.2. Commensurability and Reducibility of Lattices 10
  2. Discrete Groups of Transformations 12
  2.1. Basic Definitions and Examples 12
  2.2. Covering Sets and Fu ndamental Domains of a Discrete Group of Transformations 15
  3. Group-Theoretical Properties of Lattices in Lie Groups 17
  3.1. Finite Presentability of Lattices 17
  3.2. A Theorem of Selberg and Some of Its Consequences 18
  3.3. The Property (T) 18
  4. Intersection of Discrete Subgroups with Closed Subgroups 20
  4.1. T-Closedness of Subgroups 20
  4.2. Subgroups with Good r-Heredity 22
  4.3. Quotient Groups with Good r-Heredity 23
  4.4. T-closure 24
  5. The Space of Lattices of a Locally Compact Group 25
  5.1. Chabauty's Topology 25
  5.2. Minkowski's Lemma 25
  5.3. Mahler's Criterion 26
  6. Rigidity of Discrete Subgroups of Lie Groups 27
  6.1. Space of Homomorphisms and Deformations 27
  6.2. Rigidity and Cohomology 28
  6.3. Deformation of Uniform Subgroups 30
  7. Arithmetic Subgroups of Lie Groups 31
  7.1. Definition of an Arithmetic Subgroup 31
  7.2. When Are Arithmetic Groups Lattices (Uniform Lattices)? 32
  7.3. The Theorem of Borel and Harish-Chandra
  and the Theorem of Godement 34
  7 4. Definition of an Arithmetic Subgroup of a Lie Group 35
  8. The Borel Density Theorem 37
  8.1. The Property (S) 37
  8.2. Proof of the Density Theorem 37
  Chapter 2. Lattices in Solvable Lie Groups 39
  1. Discrete Subgroups in Abelian Lie Groups 39
  1.1. Historical Remarks 39
  1.2. Structurc of Discrete Subgroups in Simply-Connected Abelian Lic Groups 39
  1.3. Structure of Discrcte Subgroups in Arbitrary Connected Abelian Groups 40
  1.4. Use of thc Languagc of thc Thcory of AIgebraic Groups 41
  1.5. Extendability of Latticc Homomorphisms 41
  2. Lattices in Nilpotent Lic Groups 42
  2.1. Introductory Remarks and Examplcs 42
  2.2. Structurc of Latticcs in Nilpotcnt Lic Groups 43
  2.3. Latticc Homomorphisms in Nilpotcnt Lie Groups 45
  2 4. Existcncc of Latticcs in Nilpotcnt Lic Groups and Thcir Classification 46
  2.5. Lattices and Lattice Subgroups in Nilpotent Lie Groups 47
  3. Lattices in Arbitrary Solvable Lie Groups 48
  3.1. Examples of Lattices in Solvable Lie Groups of Low Dimension 48
  3.2. Topology of Solvmanifolds of Type R/T 49
  3.3. Some General Properties of Lattices in Solvable Lie Groups.50
  3.4. Mostow's Structure Theorem 51
  3.5. Wang Groups 51
  3.6. Splitting of Solvable Lie Groups 52
  3.7. Criteria for the Existence of a Lattice in a Simply-Connected Solvable Lie Group 55
  3.8. Wang Splitting and its Applications 55
  3.9. Algebraic Splitting and its Applications 58
  3.10. Linear Representability of Lattices 60
  4. Deformations and Cohomology of Lattices in Solvable Lie Groups 61
  4.1. Description of Deformations of Lattices in Simply-Connected Lie Groups 61
  4.2. On the Cohomology of Lattices in Solvable Lie Groups 63
  5. Lattices in Special Classes of Solvable Lie Groups 64
  5.1. Lattices in Solvable Lie Groups of Type (I) 64
  5.2. Lattices in Lie Groups of Type (R) 65
  5.3. Lattices in Lie Groups of Type (E) 65
  5.4. Lattices in Complex Solvable Lie Groups 66
  5.5. Solvable Lie Groups of Small Dimension, Having Lattices 66
  Chapter 3. Lattices in Semisimple Lie Groups 67
  1. General Information 68
  1.1. Red ucibility of Lattices 68
  1.2. The Density Theorem 68
  2. Reduction Theory 69
  2.1. Geometrical Language. Construction of a Reduced Basis 69
  2.2. Proof of Mahler’s Criterion 72
  2.3. The Siegel Domain 72
  3. The Theorem of Borel and Harish-Chandra(Continuation) 75
  3.1. The Case of a Torus 75
  3.2. The Semisimple Case (Siegel Domains) 77
  3.3. Proof of Godement's Theorem in the Semisimple Case 78
  4. Criteria for Uniformity of Lattices. Covolumes of Lattices 79
  4.1. Unipotent Elements in Lattices 79
  4.2. Covolumes of Lattices in Semisimple Lie Groups 80
  5. Strong Rigidity of Lattices in Semisimple Lie Groups 82
  5.1. A Theorem on Strong Rigidity 82
  5.2. Satake Compactifications of Symmetric Spaces 83
  5.3. Plan of the Proof of Mostow's Theorem 85
  6. Arithmetic Subgroups 86
  6.1. The Field Restriction Functor E 87
  6.2. Construction of Arithmetic Lattices 90
  6.3. Maximal Arithmetic Subgroups 92
  6 4. The Commensurator 94
  6.5. Normal Subgroups of Arithmetic Groups and Congruence-Subgroups 96
  6.6. The Arithmeticity Problem 96
  7. Cohomology of Lattices in Semisimple Lie Groups 97
  7.1. One-dimensional Cohomology 98
  7.2. Higher Cohomologies 100
  Chapter 4. Lattices in Lie Groups of General Type 102
  1. Bieberbach's Theorems and their Generalizations 102
  1.1. Bieberbach's Theorems 102
  1.2. Lattices in E(n) and Flat Riemannian Manifolds 106
  1.3. Generalization of the First Bieberbach Theorem 106
  2. Deformations of Lattices in Lie Groups of General Type 108
  2.1. Description of the Space of Deformations of Uniform Lattices 108
  2.2 The Levi-Mostow Decomposition for Lattices in Lie Groups of General Type 109
  3. Some Cohomological Properties of Lattices 111
  3.1. On the Cohomological Dimension of Lattices 111
  3.2. The Euler Characteristic of Lattices in Lie Groups 112
  3.3. On the Determination of Properties of Lie Groups by the Lattices in Them 113
  References 116