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Weakly Modular Graphs and Nonpositive Curvature Weakly Modular Graphs and Nonpositive Curvature J´er´emieChalopin Victor Chepoi Hiroshi Hirai Damian Osajda Author address: Laboratoire d'Informatique Fondamentale, Aix-Marseille Univer- site´ and CNRS, Faculte´ des Sciences de Luminy, F-13288 Marseille Cedex 9, France E-mail address: [email protected] Laboratoire d'Informatique Fondamentale, Aix-Marseille Univer- site´ and CNRS, Faculte´ des Sciences de Luminy, F-13288 Marseille Cedex 9, France E-mail address: [email protected] Department of Mathematical Informatics, Graduate School of In- formation Science and Technology, The University of Tokyo, Tokyo, 113-8656, Japan E-mail address: [email protected] Instytut Matematyczny, Uniwersytet Wroc lawski, pl. Grun- waldzki 2/4, 50{384 Wroc law, Poland arXiv:1409.3892v4 [math.MG] 10 Nov 2017 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich´ 8, 00-656 Warszawa, Poland E-mail address: [email protected] Contents Chapter 1. Introduction 1 1.1. Avant-propos 1 1.2. Motivation 1 1.3. Main results 5 Acknowledgements 8 Chapter 2. Preliminaries 9 2.1. Basic notions 9 2.2. Further notions 14 Chapter 3. Local-to-Global Characterization 29 3.1. Main results 29 3.2. Weakly modular graphs: proofs 31 3.3. Helly and clique-Helly graphs: proofs 38 3.4. A note on meshed graphs 50 Chapter 4. Pre-Median Graphs 53 4.1. Main results 53 4.2. Prime pre-median graphs 55 4.3. Examples 61 4.4. L1{Weakly modular graphs 66 4.5. C(G) is contractible 73 Chapter 5. Dual Polar Graphs 83 5.1. Main results 83 5.2. Proof of Theorem 5.2 85 5.3. Proof of Theorems 5.3 and 5.4 87 Chapter 6. Sweakly Modular Graphs 93 6.1. Main results 93 6.2. A lattice-theoretical characterization of swm-graphs 94 6.3. Boolean pairs and Boolean-gated sets 98 6.4. Barycentric graph G∗ 100 6.5. Thickening G∆ 105 6.6. ∆–gates and geodesic extension property 107 6.7. Normal Boolean-gated paths 108 6.8. Euclidean buildings of type Cn 109 6.9. Application I: Biautomaticity of swm-groups 114 6.10. Application II: 2{approximation to 0-extension problem on swm-graphs 117 3 4 CONTENTS Chapter 7. Orthoscheme Complexes of Modular Lattices and Semilattices 121 7.1. Main results 122 7.2. Boolean and complemented modular lattices 124 7.3. Distributive and modular lattices 125 7.4. Proof of Theorem 7.2 128 7.5. Proof of Propositions 7.5 and 7.6 129 7.6. Proof of Proposition 7.4 130 Chapter 8. Orthoscheme Complexes of Swm-Graphs 133 8.1. Main results 133 8.2. Examples 134 8.3. Preliminary results 136 8.4. l1{metrization 140 8.5. l {metrization 141 1 8.6. l2{metrization 143 8.7. K(G) is contractible 143 Chapter 9. Metric Properties of Weakly Modular Graphs 145 9.1. Quadratic isoperimetric inequality 145 9.2. Hyperbolicity 146 9.3. BFS gives a distance-preserving ordering 149 9.4. Weakly modular complex 151 Bibliography 155 Abstract This article investigates structural, geometrical, and topological characteri- zations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various \nonpositive cur- vature" and \local-to-global" properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of mod- ular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises (1{skeletons, collinearity graphs, covering graphs, domains, etc.) in several seemingly-unrelated fields of mathematics: Metric graph theory • Geometric group theory • Incidence geometries and buildings • Theoretical computer science and combinatorial optimization • We give a local-to-global characterization of weakly modular graphs and their sub- classes in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk- )Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With l1{embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even {matroids in the first case and orthoscheme complexes of gated dual polar subgraphs4 in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer Chastand's question about prime graphs for pre-median graphs. We also explore negative curvature for weakly modular graphs. 2010 Mathematics Subject Classification. 05C12, 51K05, 20F67, 90C27. Key words and phrases. Weakly modular graph, nonpositive curvature, CAT(0) space, lat- tice, building, incidence geometry, group action, combinatorial optimization. J.C. was partially supported by ANR project MACARON (anr-13-js02-0002). V.C. was partially supported by ANR projects TEOMATRO (anr-10-blan-0207) and GGAA (anr-10-blan-0116). H.H. was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 17K00029. D.O. was partially supported by Narodowe Centrum Nauki, grants no. UMO- 2012/06/A/ST1/00259 and UMO-2015/18/M/ST1/00050. 5 CHAPTER 1 Introduction 1.1. Avant-propos This article investigates structural, geometrical, and topological characteriza- tions and properties of weakly modular graphs and of cell complexes derived from them. Weakly modular graphs are defined as graphs satisfying the following two simple distance conditions (for every k > 0): Triangle condition (TC): For any vertex u and any two adjacent vertices • v; w at distance k to u, there exists a common neighbor x of v; w at distance k 1 to u. Quadrangle− condition (QC): For any vertices u; z at distance k and any • two neighbors v; w of z at distance k 1 to u, there exists a common neighbor x of v; w at distance k 2 from− u. − Weakly modular graphs have been introduced in [Che89] and [BC96] as a far- reaching common generalization of graphs studied in metric graph theory: examples are median graphs, modular graphs, Helly graphs, and bridged graphs. Our investigation is motivated by metric graph theory as well as by recent unexpected appearances of weakly modular graphs in other fields of mathematics: Geometric group theory • Incidence geometries and buildings • Theoretical computer science and combinatorial optimization • The unifying themes of our investigation are various \nonpositive-curvature" and \local-to-global" properties and characterizations of weakly modular graphs and their subclasses. 1.2. Motivation Let us mention some motivating examples and predecessors of this article (for some undefined terms, see Chapter 2). Metric graph theory. The main subject of metric graph theory is the investi- gation and structural characterization of graph classes whose standard graph-metric satisfies the main metric and convexity properties of classical metric geometries like n R endowed with l2, l1, or l {metric, hyperbolic spaces, Boolean spaces, or trees. Among such properties one1 can mention convexity of balls or of neighborhoods of convex sets, Helly property for balls, isometric and low-distortion embeddings into classical host spaces, retractions, various four-point conditions, uniqueness or existence of medians, etc.; for a survey of this theory, see [BC08]. The main classes of graphs occurring in metric graph theory are median graphs, modular graphs, Helly graphs, bridged graphs, δ{hyperbolic graphs, l1{graphs, and isometric subgraphs of hypercubes (partial cubes). Helly graphs are the graphs in 1 2 1. INTRODUCTION which the balls satisfy the Helly property. The bridged graphs are the graphs in which the neighborhoods of convex sets are convex (one of the basic properties of Euclidean convexity). Finally, median graphs can be viewed as graphs satisfying a basic property of trees and Boolean cubes (and more generally, covering graphs of distributive lattices): each triplet of vertices admits a unique median, i.e., a vertex lying simultaneously on shortest paths between any pair of vertices from the triplet. They also can be viewed as the discrete median algebras or as covering graphs of median semilattices. Similarly to median graphs, modular graphs are the graphs in which each triplet of vertices admits a (not necessarily unique) median. Later, it turned out that these graphs give rise to important cubical and sim- plicial complexes. The cubical complexes associated with median graphs (called median complexes) endowed with any of the intrinsic l1; l2; or l {metrics consti- tute geodesic metric spaces with strong properties. For example, median1 complexes equipped with the l1{metric are median metric spaces (i.e., every triplet of points has a unique median) and are isometric subspaces of l1{spaces [vdV93]. If a median complex carries the intrinsic l {metric instead, then the resulting metric space is 1 injective [MT83, vdV98]. Finally, if we impose the intrinsic l2{metric, we obtain a metric space with global non-positive curvature in the sense of Gromov [Gro87]. In fact, median graphs are exactly the 1{skeletons of CAT(0) cubical complexes [Che00]. CAT(0) cubical complexes were characterized by Gromov [Gro87] in a local-to-global combinatorial way as the simply connected cubical complexes in which the links of vertices (0{cubes) are simplicial flag complexes. Analogously to median graphs, bridged graphs have been characterized in [Che00] as the 1{skeletons of simply connected simplicial flag complexes in which the links of vertices (0{simplices) do not contain induced 4{ and 5{cycles.
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