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Groups and Spaces Kai-Uwe Bux Groups and Spaces Volume I: Groups January 28, 2007: Preliminary version, do not cite, do not distribute. [°c Kai-Uwe Bux, 2002] Preface This set of notes is based on my lectures “Important Groups” and “My Favorite Groups” which I taught at Cornell University in Spring 2002 and Spring 2003. The goal was to discuss the most important examples of infinite discrete groups in con- siderable depth and detail. Being “important” is rather a sociological concept than a mathematical one. A group is important if Mathmaticians are interested in it: Important groups are those, we talk about on dinner parties; and if you fail to know them, you will become a social outcast. Of course, any list of those groups is open to debate, and the one to follow reflects many of my personal idiosyncrasies. What is in and what is not The focus of this lecture is on particular groups. So you will not find a lot of general theorems like the following miracle. A finitely generated group has a solvable word problem if and only if it embeds into a simple subgroup of a finitely presented group. Those gadgets are sometimes mentioned, sometimes even used, but we will not bother to prove many of these results. Instead you will find results like these: • The infinite cyclic group is amenable, whereas the other finitely generated free groups are not. • Arithmetic groups are residually finite. • The group Out(Fn) has only finitely many finite subgroups up to conjugacy. The groups dicussed in this lecture can be aranged in two main blocks: On the one hand we discus free groups, Thompson’s groups, and Grigorchuk’s groups. Here the focus of the discussion is about growth and amenability. On the other hand, we will i ii meet arithmetic groups, mapping class groups of surfaces and the groups Out(Fn). Here the discussion will be centered around the construction of nice spaces, upon which these groups act. Thanks . ... to David Revelle for proofreading my crappy notes. ... to Jim Belk for helping me considerably in understanding amenability and Thompson’s group F . ... to David Benbennick for debugging the exercises and problems. ... to Ference Gerlitz for teaching me about the subgroup conjugacy problem in free groups. ... to Rotislav Grigorchuk for explaining to me the solution of the conjugacy prob- lem in his first group. Preliminary version, do not cite, do not distribute. [°c Kai-Uwe Bux, 2002] Contents I Free Groups 1 1 The Infinite Cyclic Group 2 1.1 Residual Finiteness . 3 1.2 Amenability . 3 1.2.1 Følner Sequences . 5 1.2.2 Ultralimits . 6 1.2.3 From Følner Sequences to Amenability . 9 1.3 Kazhdan’s Property (T) . 11 1.4 The Geometry of the Cayley Graph . 12 1.4.1 Ends . 14 1.4.2 Growth . 17 2 Free Groups of Finite Rank 21 2.1 Free Constructions . 22 2.2 How to Detect Free Groups . 23 2.3 Kazhdan’s Property (T) and Amenability . 27 2.3.1 Equivalent Formulations for Amenability . 28 2.4 Stallings’ Theorem . 33 2.4.1 Cohomology of Groups and the Eilenberg-Ganea Problem . 35 2.4.2 Tree Actions and Free Products . 38 2.4.3 Stallings’ Structure Theorem . 39 2.4.4 Grushko’s Theorem . 44 2.5 The Hanna Neumann Conjecture . 48 2.6 Equations in Free Groups and the Conjugacy Problem . 50 II Arithmetic Groups 53 3 SL2(Z) and the Hyperbolic Plane 54 3.1 The Symmetric Space of SL2(R)..................... 55 iii iv CONTENTS 3.2 A Fundamental Domain for SL2(Z)................... 57 3.3 The Tree of SL2(Z)............................ 61 3.4 The Conjugacy Problem . 63 3.5 Finite Quotients and Congruence Subgroups . 64 3.6 Small Cancellation Theory for Free Products . 66 3.6.1 Van Kampen Diagrams for Presentations . 66 3.6.2 Van Kampen Diagrams for Free Products . 68 4 SLn(Z) 72 5 General Arithmetic Groups 73 5.1 Preliminary Observations . 76 III Mapping Class Groups 78 6 Out(Fn) and Aut(Fn) 79 6.1 Topological Representatives for Automorphisms . 79 6.1.1 Stallings Folds . 80 6.1.2 Bounded Cancellation and the Fixed Subgroup . 84 6.2 A Generating Set for Aut(Fn)...................... 85 6.2.1 Proof of Nielsen’s Theorem . 86 6.2.2 The Homotopytype of the Complex of Forests . 88 6.3 Outer Space and its Relatives . 88 6.3.1 Categories Based on Graphs . 89 6.3.2 Marked Graphs, Labelled Graphs, and Metric Trees . 90 6.3.3 Metric Trees and R-Trees . 91 6.3.4 The Definition of Auter Space and Outer Space . 92 6.4 Proofs of Contractibility . 93 6.4.1 Proof by Continous Folding (the Trees Proof) . 93 6.4.2 Proof by Sophisticated Low-Dimensional Topology (the Spheres Proof) . 93 6.4.3 A Continuous Contracting Flow . 95 6.4.4 A Discrete Version of the Argument and an Exponential Isoperi- metric Inequality for Out(Fn) .................. 97 6.4.5 The Graphs Proof . 101 6.5 !!! FIXME !!! . 101 7 The Mapping Class Group of a Closed Surface 104 8 Braid Groups 105 Preliminary version, do not cite, do not distribute. [°c Kai-Uwe Bux, 2002] CONTENTS v IV Miscellaneous Important Groups 106 9 Coxeter Groups and Artin Groups 107 9.1 Euclidean Reflection Groups . 107 9.1.1 The Chamber Decomposition of E ................ 107 9.1.2 The Coxeter Matrix . 112 9.1.3 The Cocompact Case . 113 9.2 Coxeter Groups . 114 9.2.1 The Geometric Representation . 115 9.2.2 The Geometry of a Coxeter System . 116 9.2.3 The Deletion Condition . 122 9.2.4 The Moussong Complex . 124 9.3 Artin Groups . 128 9.3.1 The Braid Group . 128 9.3.2 General Artin Groups . 132 9.3.3 Artin Groups of Finite Type . 133 9.3.4 Right-Angled Artin Groups and the Example of M. Bestvina and N. Brady . 133 10 Grigorchuk’s First Group 136 10.1 The Infinite Binary Rooted Tree . 137 10.2 Automaton Groups and the Word Problem . 140 10.3 Burnside’s Problem . 147 10.4 Subgroup Structure . 150 10.4.1 Finite 2-groups . 153 10.4.2 Congruence Subgroups . 156 10.5 The Weight of Elements . 159 10.5.1 The Conjugacy Problem . 161 10.5.2 Intermediate Growth . 163 10.6 Amenability . 166 10.7 Presentations . 168 11 Thompson’s Group F 171 11.1 Associativity, Trees, and Homeomorphisms . 172 11.2 The Positive Monoid and Forest Diagrams . 180 11.3 Presentations for Thompson’s Group . 184 11.4 An Action of Thompson’s Group F ................... 185 11.4.1 Forest Diagrams . 187 11.4.2 Normal Forms . 190 11.4.3 A Remark on Tree Diagrams . 195 Preliminary version, do not cite, do not distribute. [°c Kai-Uwe Bux, 2002] vi CONTENTS 11.5 Subgroups and Quotients . 195 11.6 Amenability . 198 11.7 Finiteness Properties . 199 11.8 The Conjugacy Problem . 204 V Appendices 205 A Combinatorial Group Theory: Nuts and Bolts 206 A.1 Generators and Relations . 206 A.1.1 Generating Sets / Cayley Graphs . 206 A.1.2 Defining Relations / Cayley Complexes . 207 A.1.3 Van Kampen Diagrams . 212 A.1.4 Isoperimetric Inequalities . 215 A.1.5 Small Cancellation Theory . 215 A.2 Constructions . 217 A.2.1 Direct, Semidirect, and Wreath Products . 217 A.2.2 Free Products / Amalgamated Products / HNN-Extensions . 219 A.2.3 Graphs of Groups and Spaces . 219 A.3 Faithful Representations . 219 B Geometry: Nuts and Bolts 220 B.1 Metric Spaces . 220 B.2 Piecewise Geometric Complexes . 222 B.3 Group Actions . 224 B.3.1 Proper Actions . 225 B.3.2 Proper Cocompact Actions . 225 B.3.3 Abelian and Solvable Subgroups . 228 C Topology: Nuts and Bolts 229 C.1 Topological Categories . 229 C.1.1 Paracompact Spaces . 229 C.1.2 Complexes . 229 C.1.3 Posets . 232 C.2 Computing Homotopy Groups . 232 C.2.1 Fibrations and Fibre Bundles . 232 C.2.2 Combinatorial Morse Theory . 235 C.2.3 The Vietoris-Smale-Quillen Argument . 238 C.2.4 Nerves . 238 Preliminary version, do not cite, do not distribute. [°c Kai-Uwe Bux, 2002] CONTENTS vii D Finiteness Properties 240 D.1 Brown’s Criterion . 241 D.2 Applications of Brown’s Criterion . 241 D.3 The Stallings-Bieri Series . 243 D.4 Homological Finiteness Properties . 245 E Dictionary 246 E.1 Properties . 246 E.2 Prefixes . 249 E.3 Metaproperties . ..
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