From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry
About this Title
Daniel T. Wise, McGill University, Montreal, QC, Canada
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2012: Volume 117
ISBNs: 978-0-8218-8800-1 (print); 978-0-8218-9442-2 (online)
MathSciNet review: MR2986461
MSC: Primary 20F67; Secondary 20F06, 57M07
This book presents an introduction to the geometric group theory associated with nonpositively curved cube complexes. It advocates the use of cube complexes to understand the fundamental groups of hyperbolic 3-manifolds as well as many other infinite groups studied within geometric group theory.
The main goal is to outline the proof that a hyperbolic group $G$ with a quasiconvex hierarchy has a finite index subgroup that embeds in a right-angled Artin group. The supporting ingredients of the proof are sketched: the basics of nonpositively curved cube complexes, wallspaces and dual CAT(0) cube complexes, special cube complexes, the combination theorem for special cube complexes, the combination theorem for cubulated groups, cubical small-cancellation theory, and the malnormal special quotient theorem. Generalizations to relatively hyperbolic groups are discussed. Finally, applications are described towards resolving Baumslag's conjecture on the residual finiteness of one-relator groups with torsion, and to the virtual specialness and virtual fibering of certain hyperbolic 3-manifolds, including those with at least one cusp.
The text contains many figures illustrating the ideas.
Graduate students and research mathematicians interested in low-dimensional topology and geometric group theory.
Table of Contents
- Chapter 1. Overview
- Chapter 2. Nonpositively curved cube complexes
- Chapter 3. Cubical disk diagrams, hyperplanes, and convexity
- Chapter 4. Special cube complexes
- Chapter 5. Virtual specialness of malnormal amalgams
- Chapter 6. Wallspaces and their dual cube complexes
- Chapter 7. Finiteness properties of the dual cube complex
- Chapter 8. Cubulating malnormal graphs of cubulated groups
- Chapter 9. Cubical small cancellation theory
- Chapter 10. Walls in cubical small-cancellation theory
- Chapter 11. Annular diagrams
- Chapter 12. Virtually special quotients
- Chapter 13. Hyperbolicity and quasiconvexity detection
- Chapter 14. Hyperbolic groups with a quasiconvex hierachy
- Chapter 15. The relatively hyperbolic setting
- Chapter 16. Applications