The Structure of Groups with a Quasiconvex Hierarchy

by Daniel T. Wise

Overview

This monograph on the applications of cube complexes constitutes a breakthrough in the fields of geometric group theory and 3-manifold topology. Many fundamental new ideas and methodologies are presented here for the first time, including a cubical small-cancellation theory that generalizes ideas from the 1960s, a version of Dehn Filling that functions in the category of special cube complexes, and a variety of results about right-angled Artin groups. The book culminates by establishing a remarkable theorem about the nature of hyperbolic groups that are constructible as amalgams.

The applications described here include the virtual fibering of cusped hyperbolic 3-manifolds and the resolution of Baumslag's conjecture on the residual finiteness of one-relator groups with torsion. Most importantly, this work establishes a cubical program for resolving Thurston's conjectures on hyperbolic 3-manifolds, and validates this program in significant cases. Illustrated with more than 150 color figures, this book will interest graduate students and researchers working in geometry, algebra, and topology.

Product Details

ISBN-13:	9780691170459
Publisher:	Princeton University Press
Publication date:	05/04/2021
Series:	Annals of Mathematics Studies , #209
Pages:	376
Product dimensions:	6.12(w) x 9.25(h) x (d)

About the Author

Daniel T. Wise is James McGill Professor in the Department of Mathematics and Statistics at McGill University. His previous book is From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry.

Table of Contents

Acknowledgments xi

1 Introduction 1

2 CAT(0) Cube Complexes 3

2.a Basic Definitions 3

2.b Right-Angled Artin Groups 3

2.c Hyperplanes in CAT(0) Cube Complexes 4

2.d Geodesics and the Metric 4

2.e Properties of Minimal Area Cubical Disk Diagrams 5

2.f Convexity 12

2.g Hyperplanes and Their Carriers 13

2.h Splaying and Rectangles 17

2.i Annuli 20

2.j Annular Diagrams and Mamormality 22

2.k Convex Cores 24

2.l Supereonvexity 27

3 Cubical Small-Cancellation Theory 30

3.a Introduction 30

3.b Cubical Presentations 31

3.c Pieces 33

3.d Some Small-Cancellation Conditions to Bear in Mind 37

3.e Disk Diagrams and Reduced Disk Diagrams 37

3.f Rectified Disk Diagrams 44

3.g Gauss-Bonnet Theorem 50

3.h Assigning the Angles 51

3.i Nonpositive Curvature of Shards 54

3.j Tables of Small Shards 58

3.k Nonpositive Curvature of Cone-Cells via Small-Cancellation 58

3.l Internal Cone-Cells That Do Not Self-Collide 64

3.m More General Small-Cancellation Conditions and Involved Justification 67

3.n Informal Discussion of the Limits of the Theory 69

3.o Nonpositively Curved Angling Rules 70

3.p Positive Curvature along Boundary 71

3.q Ladder Theorem 72

3.r Trichotomy for Reduced Diagrams 76

3.s Examples 76

3.t Examples Arising from Special Cube Complexes 78

3.u Graded Small-Cancellation 80

3.v Some Graded Examples 81

3.w Graded Metric Small-Cancellation 82

3.x Missing Shells and Injectivity 86

3.y Short Innerpaths and Quasiconvexity 89

4 Torsion and Hyperbolicity 94

4.a Cones Embed 94

4.b Torsion 94

4.c Hyperbolicity 97

5 New Walls and the B(6) Condition 99

5.a Introduction 99

5.b Total Defects of Paths in Cones 99

5.c Generalization of the B(6) Condition 100

5.d Cyclic Quotients and the B(6) Condition 102

5.e Embedding Properties of the Cones and Hyperplane Carriers 103

5.f Defining Immersed Walls in X^* 109

5.g No Inversions 120

5.h Carriers and Quasiconvexity 121

5.i Bigons 130

5.j Square Cones 133

5.k 1-Dimensional Linear Separation 136

5.l Obtaining Proper Actions on the Dual 138

5.m Codimension-1 Subgroup Preserved 147

5.n Elliptic Annuli 148

5.o Annular Diagrams and the B(8) Condition 151

5.p Doubly Collared Annular Diagrams 157

5.q Malnormality of Wall Stabilizers 161

5.r Artin Groups 164

6 Special Cube Complexes 166

6.a Immersed Hyperplanes 166

6.b Hyperplane Definition of Special Cube Complex 166

6.c Right-Angled Artin Group Characterization 167

6.d Canonical Completion and Retraction 168

6.e Double Cosets and Virtual Specialness 171

6.f Extensions of Quasiconvex Codimension-1 Subgroups 171

6.g The Malnormal Combination Theorem 180

7 Cubulations 181

7.a Wallspaces 181

7.b Sageev's Construction 182

7.c Finiteness Properties of the Dual Cube Complex 183

7.d Virtually Cubulated 187

7.e Sparse Complexes 189

7.f Useful Subwallspaces 203

7.g Cubulating Amalgams 215

8 Malnormality and Fiber-Products 218

8.a Height and Virtual Almost Malnormality 218

8.b Fiber-Products 220

8.c Graded Systems 223

9 Splicing Walls 225

9.a Finite Cover That Is a Wallspace 225

9.b Preservation of S mall-Cancellation and Obtaining Wall Convexity 226

9.c Obtaining the Separation Properties for Pseudographs 228

10 Cutting X^* 232

10.a Hierarchies of Cubical Presentations 232

10.b Inflations 233

10.c Some Persistent Properties 235

10.d Additional Splitting along Conepoints 237

11 Hierarchies 242

12 Virtually Special Quotient Theorem 246

12.a Malnormal Special Quotient Theorem 246

12.b Proof of the Special Quotient Theorem 249

12.c Adding Higher Grade Relators 251

12.d Controlling Intersections in Quotient 255

13 Amalgams of Virtually Special Groups 266

13.a Virtually Special Amalgams 266

14 Large Fillings Are Hyperbolic and Preservation of Quasiconvexity 272

14.a Hyperbolic Fillings 272

14.b Quasiconvex Image 276

15 Relatively Hyperbolic Case 281

15.a Introduction 281

15.b Parabolic Fillings That Are Virtually Special 283

15.c Separability for Relatively Hyperbolic Hierarchies 286

15.d Residually Verifying the Double Coset Criterion 287

15.e Relative Malnormality and Separability 291

15.f The Hierarchy in the Relatively Hyperbolic Setting 292

16 Largeness and Omnipotence 304

16.a Virtual Separation and Largeness 304

16.b Omnipotence 306

17 Hyperbolic 3-Manifolds with a Geometrically Finite Incompressible Surface 310

17.a Some Background on 3-Manifolds 311

17.b Aparabolic Hierarchy 313

17.c Virtual Specialness of Hyperbolic 3-Manifolds with Boundary 314

17.d Cutting All Tori with First Surface 316

18 Limit Groups and Abelian Hierarchies 321

18.a Limit Groups 321

18.b Abelian Hierarchies 325

19 Application Towards One-Relator Groups 333

19.a Overview 333

19.b The Magnus-Moldavanskii Hierarchy 334

19.c Quasieonvexity Using the Strengthened Spelling Theorem 337

19.d Staggered 2-Complex with Torsion 340

20 Problems 343

References 345

Index 353