Geometric Group Theory: An Introduction (Universitext)

Main subject category: • Geometric group theory • Group theory • GeometryInspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology.Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability.This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises. About this book Contents 1 Introduction Part I Groups 2 Generating groups 2.1 Review of the category of groups 2.1.1 Abstract groups: axioms 2.1.2 Concrete groups: automorphism groups 2.1.3 Normal subgroups and quotients 2.2 Groups via generators and relations 2.2.1 Generating sets of groups 2.2.2 Free groups 2.2.3 Generators and relations 2.2.4 Finitely presented groups 2.3 New groups out of old 2.3.1 Products and extensions 2.3.2 Free products and amalgamated free products 2.E Exercises Part II Groups → Geometry 3 Cayley graphs 3.1 Review of graph notation 3.2 Cayley graphs 3.3 Cayley graphs of free groups 3.3.1 Free groups and reduced words 3.3.2 Free groups → trees 3.3.3 Trees → free groups 3.E Exercises 4 Group actions 4.1 Review of group actions 4.1.1 Free actions 4.1.2 Orbits and stabilisers 4.1.3 Application: Counting via group actions 4.1.4 Transitive actions 4.2 Free groups and actions on trees 4.2.1 Spanning trees for group actions 4.2.2 Reconstructing a Cayley tree 4.2.3 Application: Subgroups of free groups are free 4.3 The ping-pong lemma 4.4 Free subgroups of matrix groups 4.4.1 Application: The group SL(2,Z) is virtually free 4.4.2 Application: Regular graphs of large girth 4.4.3 Application: The Tits alternative 4.E Exercises 5 Quasi-isometry 5.1 Quasi-isometry types of metric spaces 5.2 Quasi-isometry types of groups 5.2.1 First examples 5.3 Quasi-geodesics and quasi-geodesic spaces 5.3.1 (Quasi-)Geodesic spaces 5.3.2 Geodesification via geometric realisation of graphs 5.4 The Švarc–Milnor lemma 5.4.1 Application: (Weak) commensurability 5.4.2 Application: Geometric structures on manifolds 5.5 The dynamic criterion for quasi-isometry 5.5.1 Application: Comparing uniform lattices 5.6 Quasi-isometry invariants 5.6.1 Quasi-isometry invariants 5.6.2 Geometric properties of groups and rigidity 5.6.3 Functorial quasi-isometry invariants 5.E Exercises Part III Geometry of groups 6 Growth types of groups 6.1 Growth functions of finitely generated groups 6.2 Growth types of groups 6.2.1 Growth types 6.2.2 Growth types and quasi-isometry 6.2.3 Application: Volume growth of manifolds 6.3 Groups of polynomial growth 6.3.1 Nilpotent groups 6.3.2 Growth of nilpotent groups 6.3.3 Polynomial growth implies virtual nilpotence 6.3.4 Application: Virtual nilpotence is geometric 6.3.5 More on polynomial growth 6.3.6 Quasi-isometry rigidity of free Abelian groups 6.3.7 Application: Expanding maps of manifolds 6.4 Groups of uniform exponential growth 6.4.1 Uniform exponential growth 6.4.2 Uniform uniform exponential growth 6.4.3 The uniform Tits alternative 6.4.4 Application: The Lehmer conjecture 6.E Exercises 7 Hyperbolic groups 7.1 Classical curvature, intuitively 7.1.1 Curvature of plane curves 7.1.2 Curvature of surfaces in R3 7.2 (Quasi-)Hyperbolic spaces 7.2.1 Hyperbolic spaces 7.2.2 Quasi-hyperbolic spaces 7.2.3 Quasi-geodesics in hyperbolic spaces 7.2.4 Hyperbolic graphs 7.3 Hyperbolic groups 7.4 The word problem in hyperbolic groups 7.4.1 Application: "Solving'' the word problem 7.5 Elements of infinite order in hyperbolic groups 7.5.1 Existence 7.5.2 Centralisers 7.5.3 Quasi-convexity 7.5.4 Application: Products and negative curvature 7.6 Non-positively curved groups 7.E Exercises 8 Ends and boundaries 8.1 Geometry at infinity 8.2 Ends 8.2.1 Ends of geodesic spaces 8.2.2 Ends of quasi-geodesic spaces 8.2.3 Ends of groups 8.3 The Gromov boundary 8.3.1 The Gromov boundary of quasi-geodesic spaces 8.3.2 The Gromov boundary of hyperbolic spaces 8.3.3 The Gromov boundary of groups 8.3.4 Application: Free subgroups of hyperbolic groups 8.4 Application: Mostow rigidity 8.E Exercises 9 Amenable groups 9.1 Amenability via means 9.1.1 First examples of amenable groups 9.1.2 Inheritance properties 9.2 Further characterisations of amenability 9.2.1 Følner sequences 9.2.2 Paradoxical decompositions 9.2.3 Application: The Banach–Tarski paradox 9.2.4 (Co)Homological characterisations of amenability 9.3 Quasi-isometry invariance of amenability 9.4 Quasi-isometry vs. bilipschitz equivalence 9.E Exercises Part IV Reference material A Appendix A.1 The fundamental group A.1.1 Construction and examples A.1.2 Covering theory A.2 Group (co)homology A.2.1 Construction A.2.2 Applications A.3 The hyperbolic plane A.3.1 Construction of the hyperbolic plane A.3.2 Length of curves A.3.3 Symmetry and geodesics A.3.4 Hyperbolic triangles A.3.5 Curvature A.3.6 Other models A.4 An invitation to programming Bibliography Index of notation Index Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology. Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability. This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.-- Provided by publisher