Weakly Modular Graphs and Nonpositive Curvature.

Cover
Title page
Chapter 1. Introduction
1.1. Avant-propos
1.2. Motivation
1.3. Main results
Acknowledgments
Chapter 2. Preliminaries
2.1. Basic notions
2.2. Further notions
Chapter 3. Local-to-Global Characterization
3.1. Main results
3.2. Weakly modular graphs: proofs
3.3. Helly and clique-Helly graphs: proofs
3.4. A note on meshed graphs
Chapter 4. Pre-Median Graphs
4.1. Main results
4.2. Prime pre-median graphs
4.3. Examples
4.4. ₁-Weakly modular graphs
4.5. ( ) is contractible
Chapter 5. Dual Polar Graphs
5.1. Main results
5.2. Proof of Theorem 5.2
5.3. Proof of Theorems 5.3 and 5.4
Chapter 6. Sweakly Modular Graphs
6.1. Main results
6.2. A lattice-theoretical characterization of swm-graphs
6.3. Boolean pairs and Boolean-gated sets
6.4. Barycentric graph *
6.5. Thickening ^{Δ}
6.6. Δ-gates and geodesic extension property
6.7. Normal Boolean-gated paths
6.8. Euclidean buildings of type C_{ }
6.9. Application I: Biautomaticity of swm-groups
6.10. Application II: 2-approximation to 0-extension problem on swm-graphs
Chapter 7. Orthoscheme Complexes of Modular Lattices and Semilattices
7.1. Main results
7.2. Boolean and complemented modular lattices
7.3. Distributive and modular lattices
7.4. Proof of Theorem 7.2
7.5. Proof of Propositions 7.5 and 7.6
7.6. Proof of Proposition 7.4
Chapter 8. Orthoscheme Complexes of Swm-Graphs
8.1. Main results
8.2. Examples
8.3. Preliminary results
8.4. ₁-metrization
8.5. _{∞}-metrization
8.6. ₂-metrization
8.7. ( ) is contractible
Chapter 9. Metric Properties of Weakly Modular Graphs
9.1. Quadratic isoperimetric inequality
9.2. Hyperbolicity
9.3. BFS gives a distance-preserving ordering
9.4. Weakly modular complex
Bibliography.

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