Geometric and Cohomological Methods in Group Theory.
Bridson, Martin R.
Geometric and Cohomological Methods in Group Theory. - Cambridge : Cambridge University Press, 2009. - 1 online resource (332 pages) - London Mathematical Society Lecture Note Series, 358 ; v. 358 . - London Mathematical Society Lecture Note Series, 358. .
8.1 Measure Equivalence and Quasi-Isometry.
Includes bibliographical references.
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.
An extended tour through a selection of the most important trends in modern geometric group theory.
9781139116763 1139116762 9781139127424 (electronic bk.) 113912742X (electronic bk.) 9781139114592 (electronic bk.) 113911459X (electronic bk.) 9781139107099 (ebook) 1139107097 (pbk.) (pbk.)
Geometric group theory--Congresses.
Homology theory--Congresses.
Théorie géométrique des groupes--Congrès.
Homologie--Congrès.
MATHEMATICS--Group Theory.
Geometric group theory.
Homology theory.
Electronic books.
Conference papers and proceedings.
QA183 .G43 2009
512.2 512/.2
Geometric and Cohomological Methods in Group Theory. - Cambridge : Cambridge University Press, 2009. - 1 online resource (332 pages) - London Mathematical Society Lecture Note Series, 358 ; v. 358 . - London Mathematical Society Lecture Note Series, 358. .
8.1 Measure Equivalence and Quasi-Isometry.
Includes bibliographical references.
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.
An extended tour through a selection of the most important trends in modern geometric group theory.
9781139116763 1139116762 9781139127424 (electronic bk.) 113912742X (electronic bk.) 9781139114592 (electronic bk.) 113911459X (electronic bk.) 9781139107099 (ebook) 1139107097 (pbk.) (pbk.)
Geometric group theory--Congresses.
Homology theory--Congresses.
Théorie géométrique des groupes--Congrès.
Homologie--Congrès.
MATHEMATICS--Group Theory.
Geometric group theory.
Homology theory.
Electronic books.
Conference papers and proceedings.
QA183 .G43 2009
512.2 512/.2