Discrete Subgroups of Semisimple Lie Groups

ISBN-13: 9783642057212 / Angielski / Miękka / 2010 / 390 str.

Gregori A. Margulis

Opis

A detailed treatment of the geometric aspects of discrete groups was carried out by Raghunathan in his book "Discrete subgroups of Lie Groups" which appeared in 1972. In particular he covered the theory of lattices in nilpotent and solvable Lie groups, results of Mal'cev and Mostow, and proved the Borel density theorem and local rigidity theorem ofSelberg-Weil. He also included some results on unipotent elements of discrete subgroups as well as on the structure of fundamental domains. The chapters concerning discrete subgroups of semi- simple Lie groups are essentially concerned with results which were obtained in the 1960's. The present book is devoted to lattices, i.e. discrete subgroups of finite covolume, in semi-simple Lie groups. By "Lie groups" we not only mean real Lie groups, but also the sets of k-rational points of algebraic groups over local fields k and their direct products. Our results can be applied to the theory of algebraic groups over global fields. For example, we prove what is in some sense the best possible classification of "abstract" homomorphisms of semi-simple algebraic group over global fields.

Spis treści

1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1. Algebraic Groups Over Arbitrary Fields.- 2. Algebraic Groups Over Local Fields.- 3. Arithmetic Groups.- 4. Measure Theory and Ergodic Theory.- 5. Unitary Representations and Amenable Groups.- II. Density and Ergodicity Theorems.- 1. Iterations of Linear Transformations.- 2. Density Theorems for Subgroups with Property (S)I.- 3. The Generalized Mautner Lemma and the Lebesgue Spectrum.- 4. Density Theorems for Subgroups with Property (S)II.- 5. Non-Discrete Closed Subgroups of Finite Covolume.- 6. Density of Projections and the Strong Approximation Theorem.- 7. Ergodicity of Actions on Quotient Spaces.- III. Property (T).- 1. Representations Which Are Isolated from the Trivial One-Dimensional Representation.- 2. Property (T) and Some of Its Consequences. Relationship Between Property (T) for Groups and for Their Subgroups.- 3. Property (T) and Decompositions of Groups into Amalgams.- 4. Property (R).- 5. Semisimple Groups with Property (T).- 6. Relationship Between the Structure of Closed Subgroups and Property (T) of Normal Subgroups.- IV. Factor Groups of Discrete Subgroups.- 1. b-metrics, Vitali’s Covering Theorem and the Density Point Theorem.- 2. Invariant Algebras of Measurable Sets.- 3. Amenable Factor Groups of Lattices Lying in Direct Products.- 4. Finiteness of Factor Groups of Discrete Subgroups.- V. Characteristic Maps.- 1. Auxiliary Assertions.- 2. The Multiplicative Ergodic Theorem.- 3. Definition and Fundamental Properties of Characteristic Maps.- 4. Effective Pairs.- 5. Essential Pairs.- VI. Discrete Subgroups and Boundary Theory.- 1. Proximal G-Spaces and Boundaries.- 2. ?-Boundaries.- 3. Projective G-Spaces.- 4. Equivariant Measurable Maps to Algebraic Varieties.- VII. Rigidity.- 1. Auxiliary Assertions.- 2. Cocycles on G-Spaces.- 3. Finite-Dimensional Invariant Subspaces.- 4. Equivariant Measurable Maps and Continuous Extensions of Representations.- 5. Superrigidity (Continuous Extensions of Homomorphisms of Discrete Subgroups to Algebraic Groups Over Local Fields).- 6. Homomorphisms of Discrete Subgroups to Algebraic Groups Over Arbitrary Fields.- 7. Strong Rigidity (Continuous Extensions of Isomorphisms of Discrete Subgroups).- 8. Rigidity of Ergodic Actions of Semisimple Groups.- VIII. Normal Subgroups and “Abstract” Homomorphisms of Semisimple Algebraic Groups Over Global Fields.- 1. Some Properties of Fundamental Domains for S-Arithmetic Subgroups.- 2. Finiteness of Factor Groups of S-Arithmetic Subgroups.- 3. Homomorphisms of S-Arithmetic Subgroups to Algebraic Groups.- IX. Arithmeticity.- 1. Statement of the Arithmeticity Theorems.- 2. Proof of the Arithmeticity Theorems.- 3. Finite Generation of Lattices.- 4. Consequences of the Arithmeticity Theorems I.- 5. Consequences of the Arithmeticity Theorems II.- 6. Arithmeticity, Volume of Quotient Spaces, Finiteness of Factor Groups, and Superrigidity of Lattices in Semisimple Lie Groups.- 7. Applications to the Theory of Symmetric Spaces and Theory of Complex Manifolds.- Appendices.- A. Proof of the Multiplicative Ergodic Theorem.- B. Free Discrete Subgroups of Linear Groups.- C. Examples of Non-Arithmetic Lattices.- Historical and Bibliographical Notes.- References.