**CBMS Regional Conference Series in Mathematics**

Volume: 117;
2012;
141 pp;
Softcover

MSC: Primary 20; 57;

Print ISBN: 978-0-8218-8800-1

Product Code: CBMS/117

List Price: $41.00

Individual Price: $32.80

**Electronic ISBN: 978-0-8218-9442-2
Product Code: CBMS/117.E**

List Price: $41.00

AMS Member Price: $32.80

MAA Member Price: $36.90

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#### Supplemental Materials

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

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*Daniel T. Wise*

A co-publication of the AMS and CBMS

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.

#### Readership

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

#### Table of Contents

# Table of Contents

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

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Acknowledgement xi12 free
- Note to reader xiii14 free
- Overview 116 free
- Nonpositively curved cube complexes 722
- Cubical disk diagrams, hyperplanes, and convexity 1530
- Special cube complexes 3146
- Virtual specialness of malnormal amalgams 4358
- Wallspaces and their dual cube complexes 5368
- Finiteness properties of the dual cube complex 6176
- Cubulating malnormal graphs of cubulated groups 6984
- Cubical small cancellation theory 7792
- Walls in cubical small-cancellation theory 95110
- Annular diagrams 101116
- Virtually special quotients 107122
- Hyperbolicity and quasiconvexity detection 115130
- Hyperbolic groups with a quasiconvex hierachy 121136
- The relatively hyperbolic setting 125140
- Applications 129144
- Bibliography 135150
- Index of notation and defined terms 139154 free
- Back Cover Back Cover1160