Title Geometric and Cohomological Methods in Group Theory

Published Cambridge : Cambridge University Press, 2009

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Description 1 online resource (332 pages) Series London Mathematical Society Lecture Note Series, 358 ; v. 358 London Mathematical Society Lecture Note Series, 358 Contents Cover; Title; Copyright; Contents; Preface; List of Participants; Notes on Sela's work: Limit groups and Makanin-Razborov diagrams; Contents; 1 The Main Theorem; 1.1 Introduction; 1.2 Basic properties of limit groups; 1.3 Modular groups and the statement of the main theorem; 1.4 Makanin-Razborov diagrams; 1.5 Abelian subgroups of limit groups; 1.6 Constructible limit groups; 2 The Main Proposition; 3 Review: Measured laminations and R-tree; 3.1 Laminations; 3.2 Dual trees; 3.3 The structure theorem; 3.4 Spaces of trees; 4 Proof of the Main Proposition; 5 Review: JSJ-theory 6 Limit groups are CLG's7 A more geometric approach; References; Solutions to Bestvina & Feighn's exercises on limit groups; 1 Definitions and elementary properties; 1.1?-residually free groups; 1.2 Limit groups; 1.3 Negative examples; 2 Embeddings in real algebraic groups; 3 GADs for limit groups; 4 Constructible Limit Groups; 4.1 CLGs are CSA; 4.2 Abelian subgroups; 4.3 Heredity; 4.4 Coherence; 4.5 Finite K(G, 1); 4.6 Principal cyclic splittings; 4.7 A criterion in free groups; 4.8 CLGs are limit groups; 5 The Shortening Argument; 5.1 Preliminary ideas; 5.2 The abelian part 5.3 The surface part5.4 The simplicial part; 6 Bestvina and Feighn's geometric approach; 6.1 The space of laminations; 6.2 Matching resolutions in the limit; 6.3 Finding kernel elements carried by leaves; 6.4 Examples of limit groups; References; L2 Invariants from the algebraic point of view; 0 Introduction; Contents; 1 Group von Neumann Algebras; 1.1 The Definition of the Group von Neumann Algebra; 1.2 Ring Theoretic Properties of the Group von Neumann Algebra; 1.3 Dimension Theory over the Group von Neumann Algebra; 2 Definition and Basic Properties of L2-Betti Numbers 2.1 The Definition of L2-Betti Numbers2.2 Basic Properties of L2-Betti Numbers; 2.3 Comparison with Other Definitions; 2.4 L2-Euler Characteristic; 3 Computations of L2-Betti Numbers; 3.1 Abelian Groups; 3.2 Finite Coverings; 3.3 Surfaces; 3.4 Three-Dimensional Manifolds; 3.5 Symmetric Spaces; 3.6 Spaces with S1 Action; 3.7 Mapping Tori; 3.8 Fibrations; 4 The Atiyah Conjecture; 4.1 Reformulations of the Atiyah Conjecture; 4.2 The Ring Theoretic Version of the Atiyah Conjecture; 4.3 The Atiyah Conjecture for Torsion-Free Groups; 4.4 The Atiyah Conjecture Implies the Kaplanski Conjecture 4.5 The Status of the Atiyah Conjecture4.6 Groups Without Bound on the Order of Its Finite Subgroups; 5 Flatness Properties of the Group von Neumann Algebra; 6 Applications to Group Theory; 6.1 L2-Betti Numbers of Groups; 6.2 Vanishing of L2-Betti Numbers of Groups; 6.3 L2-Betti Numbers of Some Specific Groups; 6.4 Deficiency and L2-Betti Numbers of Groups; 7 G- and K-Theory; 7.1 The K0- group of a Group von Neumann Algebra; 7.2 The K1- and L-groups of a Group von Neumann Algebra; 7.3 Applications to G-theory of Group Rings; 7.4 Applications to the Whitehead Group; 8 Measurable Group Theory Summary An extended tour through a selection of the most important trends in modern geometric group theory Notes 8.1 Measure Equivalence and Quasi-Isometry Bibliography Includes bibliographical references Notes Print version record Subject Geometric group theory -- Congresses Homology theory -- Congresses MATHEMATICS -- Group Theory. Geometric group theory Homology theory Genre/Form Conference papers and proceedings Form Electronic book Author Kropholler, Peter H Leary, Ian J. LC no. 2010275265 ISBN 9781139116763 1139116762 9781139127424 113912742X 9781139114592 113911459X 9781139107099 1139107097

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