Contents -- Preface -- Chapter 1 Introduction History and Outline -- 1.1 Lorentz manifolds and relativity -- 1.2 Symmetries of Lorentz manifolds -- 1.3 Outline of succeeding chapters -- 1.4 Notation -- 1.5 Acknowledgements -- Chapter 2 Basic Results and Definitions -- 2.1 Some set-theoretic notions -- 2.2 Some group-theoretic notions -- 2.3 Some topological notions -- 2.4 Some notions from linear algebra -- 2.5 Matrix concentration lemmas -- 2.6 First results on expansive sequences -- 2.7 Topological groups -- 2.8 Discrete groups -- 2.9 Proper actions -- 2.10 Bilinear and quadratic forms -- 2.11 Root systems -- 2.12 Minkowski forms - basic definitions -- Chapter 3 Basic Differential Topology -- 3.1 Some differential topological notions -- 3.2 Inheritability of continuity and smoothness to leanike submanifolds -- 3.3 Definition of prefoliation and foliation -- 3.4 Preliminary results to the Frobenius Theorem -- 3.5 Uniqueness in the Frobenius Theorem -- 3.6 Passage from local to global in the Frobenius Theorem -- 3.7 The Frobenius Theorem -- 3.8 Potential submersions -- 3.9 Lorentz metrics - basic definitions -- Chapter 4 Basic Lie Theoretic Results -- 4.1 Some Lie theoretic definitions and notation -- 4.2 Dynamical consequences of the Frobenius Theorem -- 4.3 exp Ad and ad -- 4.4 The Lie group Lie algebra correspondence -- 4.5 Some facts about Lie subgroups -- 4.6 The Lie algebra of [AB] -- 4.7 Lie groups and Lie algebras from bilinear and quadratic forms -- 4.8 Abelian Lie groups -- 4.9 Miscellaneous results -- 4.10 Generalities on semisimple groups and algebras -- 4.11 Real Jordan decomposition -- 4.12 Consequences of results on real Jordan decomposition -- 4.13 Generalities on algebraic groups -- 4.14 Generalities on nilpotent groups and algebras -- 4.15 Generalities on the nilradical.

4.16 Relationships between representation theories -- Chapter 5 More Lie Theory -- 5.1 Connection-preserving diffeomorphisms form a Lie group -- 5.2 The isometry group of a pseudoRiemannian manifold is a Lie group -- 5.3 More results on expansive sequences -- 5.4 Lie groups densely embedded in other Lie groups -- 5.5 Generalities on the Levi decomposition -- 5.6 Large normalizers and centralizers -- 5.7 Representation theory -- Chapter 6 Minkowski Linear Algebra -- 6.1 Notations for important elements and Lie subalgebras of so(Qd) -- 6.2 Linear algebra of Minkowski vector spaces -- 6.3 Basic calculations -- 6.4 Embeddings of Lorentz Lie algebras -- Chapter 7 Basic Dynamical Results -- 7.1 Kowalsky's Lemma -- 7.2 Higher jets of vector fields and metrics - notation -- 7.3 Matrix realizations of jets and calculus on jets -- 7.4 Miscellaneous results -- 7.5 A basic collection of rigidity results -- 7.6 Strongly lightlike and nontimelike vectors -- 7.7 Basic results on degenerate orbits -- 7.8 More on strongly lightlike and nontimelike vectors -- 7.9 Nonproperness and cocompact subgroups -- 7.10 Kowalsky subsets -- 7.11 Types of chaotic actions -- 7.12 Induction of actions: Definition -- 7.13 Induction of actions: Basic results -- 7.14 Riemannian dynamics -- Chapter 8 Examples of Actions on Compact Lorentz Manifolds -- 8.1 The isometry group of a compact quotient of SL2(R) -- 8.2 Twisted Heisenberg groups -- 8.3 SL2(R) twisted Heisenberg and closure -- Chapter 9 Examples of Nonproper Actions -- 9.1 Restriction-Induction -- 9.2 Examples constructed using a quadratic form in the Lie algebra -- 9.3 A nilpotent Lie group without nonproper Lorentz dynamics -- 9.4 Groups with SO(n 1) or SO(n 2) as a local direct factor -- 9.5 Groups with a normal subgroup isomorphic to R.

Chapter 10 Semisimple Groups Admitting a Nonproper Action -- 10.1 Locally free actions of SL3(R) -- 10.2 Locally faithful actions of SL3(R) -- 10.3 Locally faithful actions of Sp3(R) C R6x6 -- 10.4 Locally faithful actions of SU(2 1) -- 10.5 Kowalsky's theorems -- Chapter 11 Groups with Action on a Compact Lorentz Manifold -- 11.1 Local freeness -- 11.2 Nilpotent Lie groups -- 11.3 Solvable nonnilpotent Lie groups with Abelian nilradical -- 11.4 Solvable nonnilpotent Lie groups with nonAbelian nilradical - preliminaries -- 11.5 Solvable nonnilpotent Lie groups with nonAbelian nilradical - final results -- 11.6 Semisimple Lie groups -- 11.7 The general case -- Chapter 12 The Isometry Group of a Compact Lorentz Manifold -- 12.1 Oseledec splitting -- 12.2 Preliminaries -- 12.3 A candidate for the new Killing field -- 12.4 Final classification result -- Chapter 13 Highly Symmetric Compact Lorentz Manifolds -- 13.1 SL2(R)-actions on compact Lorentz manifolds -- 13.2 Actions of twisted Heisenberg groups on compact Lorentz manifolds - a sketch -- Chapter 14 Locally Free Orbit Nonproper Lorentz Actions -- 14.1 Preliminary result to degeneration of Adggi -- 14.2 Degeneration of Adggi -- 14.3 Locally free actions of simply connected Lie groups -- 14.4 Compact subgroups not in the radical -- 14.5 Sequences with no large Kowalsky subsets -- 14.6 Compact connected stabilizers -- Chapter 15 Orbit Nonproper Lorentz Actions -- 15.1 Preliminaries -- 15.2 A strongly vanishing element -- 15.3 A strongly lightlike element -- 15.4 Nilradical is free -- 15.5 Nilradical is not free but its center is free Part I -- 15.6 Nilradical is not free but its center is free Part II -- 15.7 The main result -- Appendix A The Borel Density Theorem -- A.l Notation and terminology -- A.2 Preliminaries -- A.3 The Borel Density Theorem.

Appendix B Tameness of Algebraic Actions in Characteristic Zero -- Bibliography -- Index.