Model Theory of Groups and Automorphism Groups (London Mathematical Society Lecture Note Series, Series Number 244)

This volume surveys recent interactions between model theory and other branches of mathematics, notably group theory. Beginning with an introductory chapter describing relevant background material, the book contains contributions from many leading international figures in this area. Topics described include automorphism groups of algebraically closed fields, the model theory of pseudo-finite fields and applications to the subgroup structure of finite Chevalley groups. Model theory of modules, and aspects of model theory of various classes of groups, including free groups are also discussed. The book also contains the first comprehensive survey of finite covers. Many new proofs and simplifications of recent results are presented and the articles contain many open problems. This book will be a suitable guide for graduate students and a useful reference for researchers working in model theory and algebra. Cover......Page 1 Title Page......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 8 Introduction......Page 10 Finite covers......Page 18 0 Outline of the Notes......Page 19 1.1 Definitions......Page 20 1.2 Examples......Page 23 1.3 Related Notions......Page 25 1.4 Topological arguments......Page 26 1.5 Kernels......Page 27 1.6 The model-theoretic context......Page 29 1.7 An overview......Page 32 2.1 Free covers......Page 33 2.2 Digraph coverings......Page 38 2.3 Coverings of two-graphs......Page 40 3.1 Split covers......Page 41 3.2 Regular covers and simple covers......Page 44 3.3 Minimal covers......Page 46 3.4 Irreducibility conditions......Page 49 3.5 Superlinked covers......Page 50 4.1 Elementary reductions......Page 51 4.2 Graphic triples and digraphs......Page 53 4.3 Strong types......Page 55 4.4 A vector space covering its projective space......Page 57 5.1 Strongly determined types......Page 58 5.2 Universal covers......Page 61 5.3 Highly homogeneous structures......Page 62 6.1 A strategy......Page 63 6.2 Symmetric expansions of symmetric extensions......Page 64 6.3 Applications of Pontriagin duality......Page 65 6.4 Derivations and Hc......Page 69 6.5 Finite covers of V(lto, 2) and [D]k......Page 72 6.6 Cohomology and two-graphs......Page 74 7.1 Dimension shifting and Shapiro's lemma......Page 75 7.2 Finiteness results......Page 79 8 Problems......Page 84 Definable subgroups of algebraic groups over pseudo-finite fields......Page 90 Groups in pseudofinite fields......Page 107 1. An example: one-dimensional matrix groups......Page 108 2. Algebraic groups......Page 110 3. Irreducible sets and connected groups......Page 112 4. Dimension in pseudofinite fields......Page 114 5. Algebraic groups in pseudofinite fields......Page 116 6. Almost simple groups......Page 118 7. Reduction at a prime......Page 120 8. A theorem on reduction......Page 123 The group of automorphisms of the complex numbers is complete......Page 127 The automorphism group of the field of field of complex numbers leaving fixed the algebraic numbers is simple......Page 132 The algebra of an age......Page 143 Elimination of inverses in groups......Page 151 Model-theoretic properties of polycyclic-by-finite groups......Page 161 Non-standard free groups......Page 170 Finitely generated subgroups of the free 7L[t]-group on two generators......Page 183 Rings of definable scalars and biendomorphism rings......Page 205 Recent results on simple first-order theories......Page 219 This volume surveys recent interactions between model theory and other branches of mathematics, notably group theory. Beginning with an introductory chapter describing relevant background material, the book contains contributionons from many leading international figures in this area. Topics described include automorphism groups of algebraically closed fields, the model theory of pseudo-finite fields and applications to the subgroup structure of finite Chevalley groups. Model theory of modules, and aspects of model theory of various classes of groups, including free groups are also discussed. The book also contains the first comprehensive survey of finite covers. Many new proofs and simplifications of recent results are presented and the articles contain many open problems. This book will be a suitable guide for graduate students and a useful reference for researchers working in model theory and algebra