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几何群论（影印版）

样章
作者：
Mladen Bestvina，Michah Sageev，Karen Vogtmann 编
定价：
169.00 元
版面字数：
700千字
开本：
特殊
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2023年
ISBN：
978-7-04-059310-5
物料号：
59310-00
出版时间：
2023-03-23
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
几何学
暂无
  • 目录
    • 前辅文
      • Mladen Bestvina, Michah Sageev, Karen Vogtmann Introduction
        • Michah Sageev CAT(0) Cube Complexes and Groups
          • Introduction
            • Lecture 1. CAT(0) cube complexes and pocsets
              • 1. The basics of NPC and CAT(0) complexes
                • 2. Hyperplanes
                  • 3. The pocset structure
                  • Lecture 2. Cubulations: from pocsets to CAT(0) cube complexes
                    • 1. Ultrafilters
                      • 2. Constructing the complex from a pocset
                        • 3. Examples of cubulations
                          • 4. Cocompactness and properness
                            • 5. Roller duality
                            • Lecture 3. Rank rigidity
                              • 1. Essential cores
                                • 2. Skewering
                                  • 3. Single skewering
                                    • 4. Flipping
                                      • 5. Double skewering
                                        • 6. Hyperplanes in sectors
                                          • 7. Proving rank rigidity
                                          • Lecture 4. Special cube complexes
                                            • 1. Subgroup separability
                                              • 2. Warmup - Stallings’ proof of Marshall Hall’s theorem
                                                • 3. Special cube complexes
                                                  • 4. Canonical completion and retraction
                                                    • 5. Application: separability of quasiconvex subgroups
                                                      • 6. Hyperbolic cube complexes are virtually special
                                                      • Bibliography
                                                      • Vincent Guirardel Geometric Small Cancellation
                                                        • Introduction
                                                          • Lecture 1. What is small cancellation about?
                                                            • 1. The basic setting
                                                              • 2. Applications of small cancellation
                                                                • 3. Geometric small cancellation
                                                                • Lecture 2. Applying the small cancellation theorem
                                                                  • 1. When the theorem does not apply
                                                                    • 2. Weak proper discontinuity
                                                                      • 3. SQ-universality
                                                                        • 4. Dehn fillings
                                                                        • Lecture 3. Rotating families
                                                                          • 1. Road-map of the proof of the small cancellation theorem
                                                                            • 2. Definitions
                                                                              • 3. Statements
                                                                                • 4. Proof of Theorem 3.4
                                                                                  • 5. Hyperbolicity of the quotient
                                                                                    • 6. Exercises
                                                                                    • Lecture 4. The cone-off
                                                                                      • 1. Presentation
                                                                                        • 2. The hyperbolic cone of a graph
                                                                                          • 3. Cone-off of a space over a family of subspaces
                                                                                          • Bibliography
                                                                                          • Pierre-Emmanuel Caprace Lectures on Proper CAT(0) Spaces and Their Isometry Groups
                                                                                            • Introduction
                                                                                              • Lecture 1. Leading examples
                                                                                                • 1. The basics
                                                                                                  • 2. The Cartan–Hadamard theorem
                                                                                                    • 3. Proper cocompact spaces
                                                                                                      • 4. Symmetric spaces
                                                                                                        • 5. Euclidean buildings
                                                                                                          • 6. Rigidity
                                                                                                            • 7. Exercises
                                                                                                            • Lecture 2. Geometric density
                                                                                                              • 1. A geometric relative of Zariski density
                                                                                                                • 2. The visual boundary
                                                                                                                  • 3. Convexity
                                                                                                                    • 4. A product decomposition theorem
                                                                                                                      • 5. Geometric density of normal subgroups
                                                                                                                        • 6. Exercises
                                                                                                                        • Lecture 3. The full isometry group
                                                                                                                          • 1. Locally compact groups
                                                                                                                            • 2. The isometry group of an irreducible space
                                                                                                                              • 3. de Rham decomposition
                                                                                                                                • 4. Exercises
                                                                                                                                • Lecture 4. Lattices
                                                                                                                                  • 1. Geometric Borel density
                                                                                                                                    • 2. Fixed points at infinity
                                                                                                                                      • 3. Boundary points with a cocompact stabiliser
                                                                                                                                        • 4. Back to rigidity
                                                                                                                                          • 5. Flats and free abelian subgroups
                                                                                                                                            • 6. Exercises
                                                                                                                                            • Bibliography
                                                                                                                                            • Michael Kapovich Lectures on Quasi-Isometric Rigidity
                                                                                                                                              • Introduction: What is Geometric Group Theory?
                                                                                                                                                • Lecture 1. Groups and spaces
                                                                                                                                                  • 1. Cayley graphs and other metric spaces
                                                                                                                                                    • 2. Quasi-isometries
                                                                                                                                                      • 3. Virtual isomorphisms and QI rigidity problem
                                                                                                                                                        • 4. Examples and non-examples of QI rigidity
                                                                                                                                                        • Lecture 2. Ultralimits and Morse lemma
                                                                                                                                                          • 1. Ultralimits of sequences in topological spaces
                                                                                                                                                            • 2. Ultralimits of sequences of metric spaces
                                                                                                                                                              • 3. Ultralimits and CAT(0) metric spaces
                                                                                                                                                                • 4. Asymptotic cones
                                                                                                                                                                  • 5. Quasi-isometries and asymptotic cones
                                                                                                                                                                    • 6. Morse lemma
                                                                                                                                                                    • Lecture 3. Boundary extension and quasi-conformal maps
                                                                                                                                                                      • 1. Boundary extension of QI maps of hyperbolic spaces
                                                                                                                                                                        • 2. Quasi-actions
                                                                                                                                                                          • 3. Conical limit points of quasi-actions
                                                                                                                                                                            • 4. Quasiconformality of the boundary extension
                                                                                                                                                                            • Lecture 4. Quasiconformal groups and Tukia’s rigidity theorem
                                                                                                                                                                              • 1. Quasiconformal groups
                                                                                                                                                                                • 2. Invariant measurable conformal structure for qc groups
                                                                                                                                                                                  • 3. Proof of Tukia’s theorem
                                                                                                                                                                                    • 4. QI rigidity for surface groups
                                                                                                                                                                                    • Appendix.
                                                                                                                                                                                      • 1. Hyperbolic space
                                                                                                                                                                                        • 2. Least volume ellipsoids
                                                                                                                                                                                          • 3. Different measures of quasiconformality
                                                                                                                                                                                          • Bibliography
                                                                                                                                                                                          • Mladen Bestvina Geometry of Outer Space
                                                                                                                                                                                            • Introduction
                                                                                                                                                                                              • Lecture 1. Outer space and its topology
                                                                                                                                                                                                • 1.1. Markings
                                                                                                                                                                                                  • 1.2. Metric
                                                                                                                                                                                                    • 1.3. Lengths of loops
                                                                                                                                                                                                      • 1.4. Fn-trees
                                                                                                                                                                                                        • 1.5. Topology and Action
                                                                                                                                                                                                          • 1.6. Thick part and spine
                                                                                                                                                                                                            • 1.7. Action of Out(Fn)
                                                                                                                                                                                                              • 1.8. Rank 2 picture
                                                                                                                                                                                                                • 1.9. Contractibility
                                                                                                                                                                                                                  • 1.10. Group theoretic consequences
                                                                                                                                                                                                                  • Lecture 2. Lipschitz metric, train tracks
                                                                                                                                                                                                                    • 2.1. Definitions
                                                                                                                                                                                                                      • 2.2. Elementary facts
                                                                                                                                                                                                                        • 2.3. Example
                                                                                                                                                                                                                          • 2.4. Tension graph, train track structure
                                                                                                                                                                                                                            • 2.5. Folding paths
                                                                                                                                                                                                                            • Lecture 3. Classification of automorphisms
                                                                                                                                                                                                                              • 3.1. Elliptic automorphisms
                                                                                                                                                                                                                                • 3.2. Hyperbolic automorphisms
                                                                                                                                                                                                                                  • 3.3. Parabolic automorphisms
                                                                                                                                                                                                                                    • 3.4. Reducible automorphisms
                                                                                                                                                                                                                                      • 3.5. Growth
                                                                                                                                                                                                                                        • 3.6. Pathologies
                                                                                                                                                                                                                                        • Lecture 4. Hyperbolic features
                                                                                                                                                                                                                                          • 4.1. Complex of free factors Fn
                                                                                                                                                                                                                                            • 4.2. The complex Sn of free factorizations
                                                                                                                                                                                                                                              • 4.3. Coarse projections
                                                                                                                                                                                                                                                • 4.4. Idea of the proof of hyperbolicity
                                                                                                                                                                                                                                                • Bibliography
                                                                                                                                                                                                                                                • Dave Witte Morris Some Arithmetic Groups that Do Not Act on the Circle
                                                                                                                                                                                                                                                  • Abstract
                                                                                                                                                                                                                                                    • Lecture 1. Left-orderable groups and a proof for SL(3, Z)
                                                                                                                                                                                                                                                      • 1A. Introduction
                                                                                                                                                                                                                                                        • 1B. Examples
                                                                                                                                                                                                                                                          • 1C. The main conjecture
                                                                                                                                                                                                                                                            • 1D. Left-invariant total orders
                                                                                                                                                                                                                                                              • 1E. SL(3, Z) does not act on the line
                                                                                                                                                                                                                                                                • 1F. Comments on other arithmetic groups
                                                                                                                                                                                                                                                                • Lecture 2. Bounded generation and a proof for SL(2, Z[α])
                                                                                                                                                                                                                                                                  • 2A. What is bounded generation?
                                                                                                                                                                                                                                                                    • 2B. Bounded generation of SL(2, Z[α])
                                                                                                                                                                                                                                                                      • 2C. Bounded orbits and a proof for SL(2, Z[α])
                                                                                                                                                                                                                                                                        • 2D. Implications for other arithmetic groups of higher rank
                                                                                                                                                                                                                                                                        • Lecture 3. What is an amenable group?
                                                                                                                                                                                                                                                                          • 3A. Ponzi schemes
                                                                                                                                                                                                                                                                            • 3B. Almost-invariant subsets
                                                                                                                                                                                                                                                                              • 3C. Average values and invariant measures
                                                                                                                                                                                                                                                                                • 3D. Examples of amenable groups
                                                                                                                                                                                                                                                                                  • 3E. Applications to actions on the circle
                                                                                                                                                                                                                                                                                  • Lecture 4. Introduction to bounded cohomology
                                                                                                                                                                                                                                                                                    • 4A. Definition
                                                                                                                                                                                                                                                                                      • 4B. Application to actions on the circle
                                                                                                                                                                                                                                                                                        • 4C. Computing H2 b (Γ;R)
                                                                                                                                                                                                                                                                                        • Appendix. Hints for the exercises
                                                                                                                                                                                                                                                                                          • Bibliography
                                                                                                                                                                                                                                                                                          • Tsachik Gelander Lectures on Lattices and Locally Symmetric Spaces
                                                                                                                                                                                                                                                                                            • Introduction
                                                                                                                                                                                                                                                                                              • Lecture 1. A brief overview on the theory of lattices
                                                                                                                                                                                                                                                                                                • 1. Few definitions and examples
                                                                                                                                                                                                                                                                                                  • 2. Lattices resemble their ambient group in many ways
                                                                                                                                                                                                                                                                                                    • 3. Some basic properties of lattices
                                                                                                                                                                                                                                                                                                      • 4. A theorem of Mostow about lattices in solvable groups
                                                                                                                                                                                                                                                                                                        • 5. Existence of lattices
                                                                                                                                                                                                                                                                                                          • 6. Arithmeticity
                                                                                                                                                                                                                                                                                                          • Lecture 2. On the Jordan–Zassenhaus–Kazhdan–Margulis theorem
                                                                                                                                                                                                                                                                                                            • 1. Zassenhaus neighborhood
                                                                                                                                                                                                                                                                                                              • 2. Jordan’s theorem
                                                                                                                                                                                                                                                                                                                • 3. Approximations by finite transitive spaces
                                                                                                                                                                                                                                                                                                                  • 4. Margulis’ lemma
                                                                                                                                                                                                                                                                                                                    • 5. Crystallographic manifolds
                                                                                                                                                                                                                                                                                                                    • Lecture 3. On the geometry of locally symmetric spaces and some
                                                                                                                                                                                                                                                                                                                      • finiteness theorems
                                                                                                                                                                                                                                                                                                                        • 1. Hyperbolic spaces
                                                                                                                                                                                                                                                                                                                          • 2. The thick–thin decomposition
                                                                                                                                                                                                                                                                                                                            • 3. Presentations of torsion free lattices
                                                                                                                                                                                                                                                                                                                              • 4. General symmetric spaces
                                                                                                                                                                                                                                                                                                                                • 5. Number of generators of lattices
                                                                                                                                                                                                                                                                                                                                • Lecture 4. Rigidity and applications
                                                                                                                                                                                                                                                                                                                                  • 1. Local rigidity
                                                                                                                                                                                                                                                                                                                                    • 2. Wang’s finiteness theorem
                                                                                                                                                                                                                                                                                                                                      • 3. Mostow’s rigidity theorem
                                                                                                                                                                                                                                                                                                                                        • 4. Superrigidity and arithmeticity
                                                                                                                                                                                                                                                                                                                                          • 5. Invariant random subgroups and the Nevo–Stuck–Zimmer theorem
                                                                                                                                                                                                                                                                                                                                          • Bibliography
                                                                                                                                                                                                                                                                                                                                          • Amie Wilkinson Lectures on Marked Length Spectrum Rigidity
                                                                                                                                                                                                                                                                                                                                            • Introduction
                                                                                                                                                                                                                                                                                                                                              • Lecture 1. Preliminaries
                                                                                                                                                                                                                                                                                                                                                • 1. Background on negatively curved surfaces
                                                                                                                                                                                                                                                                                                                                                  • 2. A key example
                                                                                                                                                                                                                                                                                                                                                    • 3. Geodesics in negative curvature
                                                                                                                                                                                                                                                                                                                                                      • 4. The geodesic flow
                                                                                                                                                                                                                                                                                                                                                      • Lecture 2. Geometry and dynamics in negative curvature
                                                                                                                                                                                                                                                                                                                                                        • 1. Busemann functions and horospheres
                                                                                                                                                                                                                                                                                                                                                          • 2. The space of geodesics and the boundary at infinity
                                                                                                                                                                                                                                                                                                                                                            • 3. The Liouville current, the cross ratio and the canonical contact form
                                                                                                                                                                                                                                                                                                                                                              • 4. Summary: a dictionary
                                                                                                                                                                                                                                                                                                                                                              • Lecture 3. The proof, Part I: A volume preserving conjugacy
                                                                                                                                                                                                                                                                                                                                                                • 1. Otal’s Proof
                                                                                                                                                                                                                                                                                                                                                                • Lecture 4. The proof, Part II: Volume preserving implies isometry
                                                                                                                                                                                                                                                                                                                                                                  • Final Comments
                                                                                                                                                                                                                                                                                                                                                                    • Bibliography
                                                                                                                                                                                                                                                                                                                                                                    • Emmanuel Breuillard Expander Graphs, Property (τ ) and Approximate Groups
                                                                                                                                                                                                                                                                                                                                                                      • Foreword
                                                                                                                                                                                                                                                                                                                                                                        • Lecture 1. Amenability and random walks
                                                                                                                                                                                                                                                                                                                                                                          • A. Amenability, Folner criterion
                                                                                                                                                                                                                                                                                                                                                                            • B. Isoperimetric inequality, edge expansion
                                                                                                                                                                                                                                                                                                                                                                              • C. Invariant means
                                                                                                                                                                                                                                                                                                                                                                                • D. Random walks on groups, the spectral radius and Kesten’s criterion
                                                                                                                                                                                                                                                                                                                                                                                  • E. Further facts and questions about growth of groups and random walks
                                                                                                                                                                                                                                                                                                                                                                                    • F. Exercise: Paradoxical decompositions, Ponzi schemes and Tarski numbers
                                                                                                                                                                                                                                                                                                                                                                                    • Lecture 2. The Tits alternative and Kazhdan’s property (T)
                                                                                                                                                                                                                                                                                                                                                                                      • A. The Tits alternative
                                                                                                                                                                                                                                                                                                                                                                                        • B. Kazhdan’s property (T)
                                                                                                                                                                                                                                                                                                                                                                                          • C. Uniformity issues in the Tits alternative, non-amenability and Kazhdan’s property (T)
                                                                                                                                                                                                                                                                                                                                                                                          • Lecture 3. Property (τ ) and expanders
                                                                                                                                                                                                                                                                                                                                                                                            • A. Expander graphs
                                                                                                                                                                                                                                                                                                                                                                                              • B. Property (τ )
                                                                                                                                                                                                                                                                                                                                                                                              • Lecture 4. Approximate groups and the Bourgain-Gamburd method
                                                                                                                                                                                                                                                                                                                                                                                                • A. Which finite groups can be turned into expanders?
                                                                                                                                                                                                                                                                                                                                                                                                  • B. The Bourgain-Gamburd method
                                                                                                                                                                                                                                                                                                                                                                                                    • C. Approximate groups
                                                                                                                                                                                                                                                                                                                                                                                                      • D. Random generators and the uniformity conjecture
                                                                                                                                                                                                                                                                                                                                                                                                        • E. Super-strong approximation
                                                                                                                                                                                                                                                                                                                                                                                                        • Appendix. The Brooks-Burger transfer
                                                                                                                                                                                                                                                                                                                                                                                                          • Bibliography
                                                                                                                                                                                                                                                                                                                                                                                                          • Martin R. Bridson Cube Complexes, Subgroups of Mapping Class Groups, and Nilpotent Genus
                                                                                                                                                                                                                                                                                                                                                                                                            • 1. Introduction
                                                                                                                                                                                                                                                                                                                                                                                                              • 2. Subgroups of mapping class groups
                                                                                                                                                                                                                                                                                                                                                                                                                • 3. Fibre products and subdirect products of free groups
                                                                                                                                                                                                                                                                                                                                                                                                                  • 4. A new level of complication
                                                                                                                                                                                                                                                                                                                                                                                                                    • 5. The nilpotent genus of a group
                                                                                                                                                                                                                                                                                                                                                                                                                      • 6. Cubes, RAAGs and CAT(0)
                                                                                                                                                                                                                                                                                                                                                                                                                        • 7. Rips, fibre products and 1-2-3
                                                                                                                                                                                                                                                                                                                                                                                                                          • 8. Examples template
                                                                                                                                                                                                                                                                                                                                                                                                                            • 9. Proofs from the template
                                                                                                                                                                                                                                                                                                                                                                                                                              • 10. The isomorphism problem for subgroups of RAAGs and Mod(S)
                                                                                                                                                                                                                                                                                                                                                                                                                                • 11. Dehn functions
                                                                                                                                                                                                                                                                                                                                                                                                                                  • Bibliography

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