Geometric and Cohomological Methods in Group Theory (London Mathematical Society Lecture Note Series, Vol. 358)

"Geometric group theory is a vibrant subject at the heart of modern mathematics. It is currently enjoying a period of rapid growth and great influence marked by a deepening of its fertile interactions with logic, analysis and large-scale geometry, and striking progress has been made on classical problems at the heart of cohomological group theory." "This volume provides the reader with a tour through a selection of the most important trends in the field, including limit groups, quasi-isometric rigidity, non-positive curvature in group theory, and L[superscript 2]-methods in geometry, topology and group theory. Major survey articles exploring recent developments in the field are supported by shorter research papers, which are written in a style that readers approaching the field for the first time will find inviting."--Jacket Cover 1 Title 5 Copyright 6 Contents 7 Preface 8 List of Participants 10 Notes on Sela’s work: Limit groups and Makanin-Razborov diagrams 13 Contents 13 1 The Main Theorem 13 1.1 Introduction 13 1.2 Basic properties of limit groups 15 1.3 Modular groups and the statement of the main theorem 16 1.4 Makanin-Razborov diagrams 19 1.5 Abelian subgroups of limit groups 19 1.6 Constructible limit groups 21 2 The Main Proposition 23 3 Review: Measured laminations and R-tree 23 3.1 Laminations 24 3.2 Dual trees 26 3.3 The structure theorem 27 3.4 Spaces of trees 28 4 Proof of the Main Proposition 30 5 Review: JSJ-theory 32 6 Limit groups are CLG's 34 7 A more geometric approach 35 References 38 Solutions to Bestvina & Feighn’s exercises on limit groups 42 1 Definitions and elementary properties 42 1.1 ω-residually free groups 42 1.2 Limit groups 43 1.3 Negative examples 44 2 Embeddings in real algebraic groups 46 3 GADs for limit groups 47 4 Constructible Limit Groups 50 4.1 CLGs are CSA 50 4.2 Abelian subgroups 52 4.3 Heredity 52 4.4 Coherence 53 4.5 Finite K(G, 1) 54 4.6 Principal cyclic splittings 54 4.7 A criterion in free groups 55 4.8 CLGs are limit groups 56 5 The Shortening Argument 61 5.1 Preliminary ideas 61 5.2 The abelian part 62 5.3 The surface part 65 5.4 The simplicial part 65 6 Bestvina and Feighn’s geometric approach 69 6.1 The space of laminations 69 6.2 Matching resolutions in the limit 70 6.3 Finding kernel elements carried by leaves 71 6.4 Examples of limit groups 73 References 74 L2 Invariants from the algebraic point of view 75 0 Introduction 75 Contents 76 1 Group von Neumann Algebras 78 1.1 The Definition of the Group von Neumann Algebra 78 1.2 Ring Theoretic Properties of the Group von Neumann Algebra 80 1.3 Dimension Theory over the Group von Neumann Algebra 81 2 Definition and Basic Properties of L2-Betti Numbers 86 2.1 The Definition of L2-Betti Numbers 87 2.2 Basic Properties of L2-Betti Numbers 89 2.3 Comparison with Other Definitions 93 2.4 L2-Euler Characteristic 94 3 Computations of L2-Betti Numbers 97 3.1 Abelian Groups 98 3.2 Finite Coverings 98 3.3 Surfaces 98 3.4 Three-Dimensional Manifolds 98 3.5 Symmetric Spaces 100 3.6 Spaces with S1 Action 101 3.7 Mapping Tori 102 3.8 Fibrations 103 4 The Atiyah Conjecture 104 4.1 Reformulations of the Atiyah Conjecture 104 4.2 The Ring Theoretic Version of the Atiyah Conjecture 105 4.3 The Atiyah Conjecture for Torsion-Free Groups 108 4.4 The Atiyah Conjecture Implies the Kaplanski Conjecture 109 4.5 The Status of the Atiyah Conjecture 109 4.6 Groups Without Bound on the Order of Its Finite Subgroups 111 5 Flatness Properties of the Group von Neumann Algebra 112 6 Applications to Group Theory 113 6.1 L2-Betti Numbers of Groups 113 6.2 Vanishing of L2-Betti Numbers of Groups 116 6.3 L2-Betti Numbers of Some Specific Groups 117 6.4 Deficiency and L2-Betti Numbers of Groups 119 7 G- and K-Theory 122 7.1 The K0- group of a Group von Neumann Algebra 122 7.2 The K1- and L-groups of a Group von Neumann Algebra 126 7.3 Applications to G-theory of Group Rings 126 7.4 Applications to the Whitehead Group 128 8 Measurable Group Theory 128 8.1 Measure Equivalence and Quasi-Isometry 129 8.2 Discrete Measured Groupoids 130 8.3 Groupoid Rings 132 8.4 L2- Betti Numbers of Standard Actions 134 8.5 Invariance of L2-Betti Numbers under Orbit Equivalence 135 9 The Singer Conjecture 136 9.1 The Singer Conjecture and the Hopf Conjecture 136 9.2 Pinching Conditions 137 9.3 The Singer Conjecture and K ̈ahler Manifolds 138 10 The Approximation Conjecture 139 11 L2-Torsion 141 11.1 The Fuglede-Kadison Determinant 141 11.2 The Determinant Conjecture 144 11.3 Definition and Basic Properties of L2-Torsion 146 11.4 Computations of L2-Torsion 149 11.5 Some Open Conjectures about L2-Torsion 150 11.6 L2 Torsion of Group Automorphisms 151 12 Novikov-Shubin Invariants 152 12.1 Definition of Novikov-Shubin Invariants 152 12.2 Basic Properties of Novikov-Shubin Invariants 154 12.3 Computations of Novikov-Shubin Invariants 155 12.4 Open conjectures about Novikov-Shubin invariants 156 13 A combinatorial approach to L2-invariants 158 14 Miscellaneous 160 References 161 Notation 169 Index 170 Constructing non-positively curved spaces and groups 174 1 Negative curvature 175 1.1 Varieties of negative curvature 175 1.2 Curvature conjecture 180 1.3 Decidability 185 1.4 Length spectrum 189 2 Algorithms 190 2.1 Algorithm in dimension 191 2.2 3-manifold results 195 2.3 Higher dimensions 199 3 Special classes of groups 201 3.1 Coxeter groups and Artin groups 201 3.2 Small cancellation groups 209 3.3 Ample twisted face-pairing groups 213 4 Combinatorial notions of curvature 214 4.1 Angles in Polytopes 214 4.2 Combinatorial Gauss-Bonnet 221 4.3 Conformal CAT(0) and sectional curvature 225 References 230 Homology and dynamics in quasi-isometric rigidity of once-punctured mapping class groups 237 1 Results and techniques in QI-rigidity 238 2 Fiber preserving quasi-isometries 245 2.1 Uniformly finite homology 246 2.2 Top dimensional supports. 249 2.3 Application to fiber bundles. 250 3 Extensions of surface groups and Mess subgroups 252 3.1 Dimension of MCG(Sg). 253 3.2 Mess subgroups 254 3.3 Model spaces and Mess cycles 255 3.4 Passage to cosets of curve stabilizers 256 3.5 New model space 257 3.6 The subgraphs Гc coarsely separate points 257 4 Dynamical techniques: extensions of surface groups by pseudo-Anosov homeomorphisms. 257 4.1 Teichmüller space and its canonical H2 bundle. 258 4.2 Proof of fibered QI-rigidity 261 4.3 Proof of Theorem 21 263 References 265 Hattori-Stallings trace and Euler characteristics for groups 268 Introduction 268 1 Review of the Bass conjecture 269 2 Euler characteristics 272 3 Stability properties of Formula (1) 276 4 Conjecture 1 and two classes of groups 279 References 281 Groups of small homological dimension and the Atiyah conjecture 284 1 Notation 284 2 Review of the Atiyah Conjecture 284 3 Results 285 4 Proofs 285 References 288 Logarithms and assembly maps on Kn(Zl[G]) 290 1 Introduction 290 2 K-theory of l-adic local fields 291 3 A logarithmic construction 292 4 Assembly maps 296 References 300 On complete resolutions 303 0 Introduction 303 1 Complete cohomology and Complete resolutions 304 2 silpG, silpG, fin. dim of ZG and Complete resolutions 309 3 Complete resolutions and free actions on finite dimensional homotopy spheres 315 References 316 Structure theory for branch groups 318 1 Introduction 318 2 Examples and motivation 319 3 The structure graph 323 3.1 The structure lattice 324 3.2 Basal subgroups and the graph B 325 4 Structure trees 327 5 The relation between the graph B and trees 329 References 330 Geometric group theory is a vibrant subject at the heart of modern mathematics. Major survey articles form the backbone of this book, providing an extended tour through a selection of the most important trends in modern geometric group theory, supported by shorter research articles on diverse topics.