viii CONTENTS Chapter 7. Finiteness properties of the dual cube complex 61 7.1. The Cubes of C: 61 7.2. The Bounded Packing Property and Finite Dimensionality: 62 7.3. Cocompactness in the Hyperbolic Case 63 7.4. Relative Cocompactness in the Relatively Hyperbolic Case 63 7.5. Properness of the G Action on C( ̃) 65 7.6. The Cut-Wall Criterion for Properness 67 Chapter 8. Cubulating Malnormal Graphs of Cubulated Groups 69 8.1. A Wallspace for an Easy Non-Hyperbolic Group 69 8.2. Extending Walls 71 8.3. Constructing Turns 72 8.4. Cubulating Malnormal Amalgams 73 Chapter 9. Cubical Small Cancellation Theory 77 9.1. Cubical Presentations 78 9.2. The Fundamental Theorem of Small-Cancellation Theory 79 9.3. Combinatorial Gauss-Bonnet Theorem 81 9.4. Greendlinger’s Lemma and the Ladder Theorem 82 9.5. Reduced Diagrams 84 9.6. Producing Examples 87 9.7. Rectified Diagrams 88 Chapter 10. Walls in Cubical Small-Cancellation Theory 95 10.1. Walls in Classical C′( 1 6 ) Small-Cancellation Complexes 95 10.2. Wallspace Cones 95 10.3. Producing Wallspace Cones 96 10.4. Walls in ̃∗ 97 10.5. Quasiconvexity of Walls in ̃∗ 98 Chapter 11. Annular Diagrams 101 11.1. Classification of Flat Annuli 101 11.2. The Doubly Collared Annulus Theorem 103 11.3. Almost Malnormality 104 Chapter 12. Virtually Special Quotients 107 12.1. The Malnormal Special Quotient Theorem 107 12.2. Case Study: F2/⟨⟨W1 n1 ,...,W nr r ⟩⟩ 109 12.3. The Special Quotient Theorem 113 Chapter 13. Hyperbolicity and Quasiconvexity Detection 115 13.1. Cubical Version of Filling Theorem 115 13.2. Persistence of Quasiconvexity 117 13.3. No Missing Shells and Quasiconvexity 117 Chapter 14. Hyperbolic groups with a quasiconvex hierarchy 121 Chapter 15. The relatively hyperbolic setting 125
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