Contents Acknowledgement xi Note to the reader xiii Chapter 1. Overview 1 1.1. Applications 4 1.2. A Scheme for Understanding Groups 5 Chapter 2. Nonpositively Curved Cube Complexes 7 2.1. Definitions 7 2.2. Some Favorite 2-Dimensional Examples 8 2.3. Right-Angled Artin Groups 12 2.4. Hyperplanes 13 Chapter 3. Cubical disk diagrams, hyperplanes, and convexity 15 3.1. Disk Diagrams 15 3.2. Properties of Hyperplanes 18 3.3. Local Isometries and Convexity 22 3.4. Background on Quasiconvexity 25 3.5. Cores, Hulls, and Superconvexity 27 Chapter 4. Special Cube Complexes 31 4.1. Hyperplane Definition of Special Cube Complex 31 4.2. Separability Criteria for Virtual Specialness 33 4.3. Canonical Completion and Retraction 35 4.4. Separability in the Hyperbolic Case 36 4.5. Wall-Injectivity and a Fundamental Commutative Diagram 39 4.6. Wall Projection Controls Retraction 40 Chapter 5. Virtual Specialness of Malnormal Amalgams 43 5.1. Specializing Malnormal Amalgams 43 5.2. Proof of the Isomorphic Elevation Lemma 48 Chapter 6. Wallspaces and their Dual Cube Complexes 53 6.1. Wallspaces 53 6.2. The Dual CAT(0) Cube Complex 53 6.3. C is CAT(0) 55 6.4. Some Examples 56 6.5. Wallspaces from Codimension-1 Subgroups 58 vii
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