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From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry

Daniel T. Wise McGill University, Montreal, QC, Canada
A co-publication of the AMS and CBMS
Available Formats:
Softcover ISBN: 978-0-8218-8800-1
Product Code: CBMS/117
141 pp
List Price: $44.00 Individual Price:$35.20
Electronic ISBN: 978-0-8218-9442-2
Product Code: CBMS/117.E
141 pp
List Price: $41.00 MAA Member Price:$36.90
AMS Member Price: $32.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$66.00
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From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry
Daniel T. Wise McGill University, Montreal, QC, Canada
A co-publication of the AMS and CBMS
Available Formats:
 Softcover ISBN: 978-0-8218-8800-1 Product Code: CBMS/117 141 pp
 List Price: $44.00 Individual Price:$35.20
 Electronic ISBN: 978-0-8218-9442-2 Product Code: CBMS/117.E 141 pp
 List Price: $41.00 MAA Member Price:$36.90 AMS Member Price: $32.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$66.00
• Book Details

CBMS Regional Conference Series in Mathematics
Volume: 1172012
MSC: Primary 20; 57;

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.

A co-publication of the AMS and CBMS.

Graduate students and research mathematicians interested in low-dimensional topology and geometric group theory.

• Chapters
• 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

• Request Review Copy
Volume: 1172012
MSC: Primary 20; 57;

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.

A co-publication of the AMS and CBMS.

Graduate students and research mathematicians interested in low-dimensional topology and geometric group theory.

• Chapters
• 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
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