Lie groups and lie algebras II : discrete subgroups of lie groups and cohomologies of lie groups and lie algebras

A systematic survey of all the basic results on the theory of discrete subgroups of Lie groups, presented in a convenient form for users. The book makes the theory accessible to a wide audience, and will be a standard reference for many years to come. Review "... As always, a most useful compendium." Mathematika 2002, Issue 93-94 I. DISCRETE SUBGROUPS OF LIE GROUPS Chapter 0. Introduction Chapter 1. Discrete Subgroups of Locally Compact Topological Groups 1. The Simplest Properties of Lattices 1.1 Definition of a Discrete Subgroup. Examples 1.2 Commensurability and Reducibility of Lattices 2. Discrete Groups of Transformations 2.1 Basic Definitions and Examples 2.2 Covering Sets and Fundamental Domains of a Discrete Group of Transformations 3. Group-Theoretical Properties of Lattices in Lie Groups 3.1 Finite Presentability of Lattices 3.2 A Theorem of Selberg and Some of Its Consequences 3.3 The Property (T) 4. Intersection of Discrete Subgroups with Closed Subgroups 4.1 P-Closedness of Subgroups 4.2 Subgroups with Good F-Heredity 4.3 Quotient Groups with Good F-Heredity 4.4 F-Closure 5. The Space of Lattices of a Locally Compact Group 5.1 Chabauty's Topology 5.2 Minimwaki?ルs Lemma 5.3 Mnhlor's Criterion 6. Rigidity of Discrete Subgroups of Lie Groups 6.1 Space of Homomorphisms and Deformations 6.2 Rigidity and Cohomology 6.3 Deformation of Uniform Subgroups 7. Arithmetic Subgroups of Lie Groups 7.1 Definition of an Arithmetic Subgroup 7.2 When Are Arithmetic Groups Lattices (Uniform Lattices)? 7.3 The Theorem of Borel and Harish-Chandra and the Theorem of Godement 8. The Borel Density Theorem 8.1 The Property (S) 8.2 Proof of the Density Theorem Chapter 2. Lattices in Solvable Lie Groups 1. Discrete Subgroups in Abelian Lie Groups 1.1 Historical Remarks 1.2 Structure of Discrete Subgroups in Simply-Connected Abelian Lie Groups 1.3 Structure of Discrete Subgroups in Arbitrary Connected Abelian Groups 1.4 Use of the Language of the Theory of Algebraic Groups 1.5 Extendability of Lattice Homomorphisms 2. Lattices in Nilpotent Lie Groups 2.1 Introductory Remarks and Examples 2.2 Structure of Lattices in Nilpotent Lie Groups 2.3 Lattice Homomorphisms in Nilpotent Lie Groups 2.4 Existence of Lattices in Nilpotent Lie Groups and Their Classification 2.5 Lattices and Lattice Subgroups in Nilpotent Lie Groups 3. Lattices in Arbitrary Solvable Lie Groups 3.1 Examples of Lattices in Solvable Lie Groups of Low Dimension 3.2 Topology of Solvmanifolds of Type 12/" 3.3 Some General Properties of Lattices in Solvable Lie Groups 3.4 Mostow's Structure Theorem 3.5 Wang Groups 3.6 Splitting of Solvable Lie Groups 3.7 Criteria for the Existence of a Lattice in a Simply-Connected Solvable Lie Group 3.8 Wang Splitting and its Applications 3.9 Algebraic Splitting and its Applications 3.10 Linear Representability of Lattices 4 Deformations and Cohomology of Lattices in Solvable Lie Groups 4.1 Description of Deformations of Lattices in Simply-Connected Lie Groups 4.2 On the Cohomology of Lattices in Solvable Lie Groups 5. Lattices in Special Classes of Solvable Lie Groups 5.1 Lattices in Solvable Lie Groups of Type (I) 5.2 Lattices in Lie Groups of Type (R) 5.3 Lattices in Lie Groups of Type (E) 5.4 Lattices in Complex Solvable Lie Groups 5.5 Solvable Lie Groups of Small Dimension, Having Lattices Chapter 3. Lattices in Semisimple Lie Groups 1. General Information 1.1 Reducibility of Lattices 1.2 The Density Theorem 2. Reduction Theory 2.1 Geometrical Language. Construction of a Reduced Basis 2.2 Proof of Mahler's Criterion 2.3 The Siegel Domain 3. The Theorem of Borel and Harish-Chandra (Continuation) 3.1 The Case of a Torus 3.2 The Semisimple Case (Siegel Domains) 3.3 Proof of Godement?ルs Theorem in the Semisimple Case 4. Criteria for Uniformity of Lattices. Covolumes of Lattices 4.1 Unipotent Elements in Lattices 4.2 Covolumes of Lattices in Semisimple Lie Groups 5. Strong Rigidity of Lattices in Semisimple Lie Groups 5.1 A Theorem on Strong Rigidity 5.2 Satake Compactifications of Symmetric Spaces 5.3 Plan of the Proof of Mostow's Theorem 6. Arithmetic Subgroups 6.1 The Field Restriction Functor 6.2 Constructions of Arithmetic Lattices 6.3 Maximal Arithmetic Subgroups 6.4 The Commensurator 6.5 Normal Subgroups of Arithmetic Groups and Congruence-Subgroups 6.6 The Arithmeticity Problem 7. Cohomology of Lattices in Semisimple Lie Groups 7.1 One-dimensional Cohomology 7.2 Higher Cohomologies Chapter 4. Lattices in Lie Groups of General Type 1. Bieberbach?ルs Theorem and their Generalizations 1.1 Bieberbach?ルs Theorems 1.2 Lattices in E(n) and Flat Riemannian Manifolds 1.3 Generalization of the First Bieberbach Theorem 2. Deformations of Lattices in Lie Groups of General Type 2.1 Description of the Space of Deformations of Uniform Lattices 2.2 The ᅣフech-Mostow Decomposition for Lattices in Lie Groups of General Type 3. Some Cohomological Properties of Lattices 3.1 On the Cohomological Dimension of Lattices 3.2 The Euler Characteristic of Lattices in Lie Groups 3.3 On the Determination of Properties of Lie Groups by the Lattices in Them References II. COHOMOLOGIES OF LIE GROUPS AND LIE ALGEBRAS Chapter 1. General Theory 1. Basic Definitions 1.1. Homology and Cohomology of Discrete Groups 1.2. Inclusion of Topologies 1.3. Lie Algebra Cohomology and Homology 1.4. Generalization: Cohomology of Semisimple Sheaves. Segal Cohomology 2. Simplest General Properties 2.1. Induced Homomorphisms and Coefficient Sequences 2.2. Poincarᅢᄅ Duality 2.3. Triviality of the Actions of Lie Groups and Lie Algebras on Their Cohomologies 3. Relations between Various Homologies and Cohomologies 3.1. Topological Cohomology of a Lie Group and the Cohomology of the Corresponding Lie Algebras 3.2. The Hochschild-Mostow Spectral Sequence 3.3. Cohomology of Lie Groups and their Discrete Subgroups 3.4. Cohomology of the Classifying Space of a Lie Group and Its Discrete Cohomology 4. The Basic Means of Computation 4.1. The Hochschild-Serre Spectral Sequence 4.2. Connection with the Induced and Coinduced Modules 4.3. Inner Grading Chapter 2. Interpretation of Cohomology and Homology of Small Dimension 1. Zero-dimensional and One-dimensional Cohomology and Homology 1.1. Zero-dimensional Cohomology and Homology 1.2. One-dimensional Cohomology and Homology 2. Extensions 2.1. Extensions of Groups and Cohomology 2.2. Extensions of Lie Algebras and Cohomology 2.3. Extension of Topological Groups and Cohomology 3. Deformations 3.1. Deformations of the Lie Algebra ?メᄁ and Hn(?メᄁ, ?メᄁ) 3.2. Versal Deformations of Lie Algebras in Other Algebraic Structures Chapter 3. Calculations 1. Finite-Dimensional Lie Algebras 1.1. The Case of Trivial Coefficients 1.2. The Case of Nontrivial Coefficients 1.3. The Bott-Kostant Theorem 2. Lie Algebras of Vector Fields 2.1. The Complete Lie Algebra of Formal Vector Fields 2.2. The Case ?ムロ = -1 2.3. Other Classical Lie Algebras of Vector Fields 2.4. Lie Algebras of Smooth Vector Fields on Manifolds 2.5. Cohomology of Diffeomorphism Groups 3. Current Algebras 3.1. Current Algebras and Groups and Gauge Groups 3.2. Cohomology of Current Algebras 3.3. Cohomology of the Lie Algebras of the Group H(E) 4. Lie Algebras of Infinite Matrices 4.1. Introduction 4.2. Definitions of Lie Algebras of Infinite Matrices 4.3. The Lie Algebras L(EndV) and L(End, V) 4.4. The Lie Algebra of Generalized Jacobian Matrices, and Lie Algebras Related to It 4.5. Generalization 4.6. The Lie Algebra of Differential Operators 5. Semi-Infinite Homology 5.1. Definition 5.2. Calculations for the Virasoro and Kac-Moody Algebras 6. Explicit Formulas for Cocycles of Groups 6.1. The Construction of Gelfand-Fuks 6.2. The Construction of Borel Comments on the Literature References AUTHOR INDEX SUBJECT INDEX

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. Since 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. Part II on Cohomologies of Lie Groups and 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 infinite-dimensional) 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.

"Various types of discrete subgroups of Lie groups arise in the theory of functions of complex variables, arithmetic, geometry, and crystallography. Since 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. The book can be useful as a reference and research guide to graduate students and researchers in different areas of mathematics and theoretical physics."--Jacket