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Gluing Constructions and Local-To-Global Results for Hyperconvex Metric Spaces

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Gluing Constructions and Local-To-Global Results for Hyperconvex Metric Spaces Research Collection Doctoral Thesis Gluing Constructions and Local-to-Global Results for Hyperconvex Metric Spaces Author(s): Miesch, Benjamin Publication Date: 2017 Permanent Link: https://doi.org/10.3929/ethz-a-010867059 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 23997 Gluing Constructions and Local-to-Global Results for Hyperconvex Metric Spaces A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH¨ (Dr. sc. ETH Z¨urich) presented by Benjamin Raphael Miesch MSc ETH Mathematik, ETH Z¨urich born May 14, 1987 citizen of Titterten BL accepted on the recommendation of Prof. Dr. Urs Lang, examiner Prof. Dr. Rafael Esp´ınola,co-examiner 2017 Soli Deo Gloria. Zusammenfassung Hyperkonvexit¨at beschreibt eine Schnitteigenschaft von B¨allen in metrischen R¨aumen und wurde 1956 von N. Aronszajn and P. Panitchpakdi fur¨ das Studi- um von Erweiterungen gleichm¨assig stetiger Abbildungen eingefuhrt.¨ In dieser Arbeit studieren wir nun verschiedene Eigenschaften von hyperkon- vexen metrischen R¨aumen. Zuerst untersuchen wir Verklebungen von solchen R¨aumen entlang isometrischer Teilmengen. Fur¨ schwach extern hyperkonvexe Klebmengen k¨onnen wir eine notwendige und hinreichende Bedingung angeben, so dass der resultierende Raum wieder hyperkonvex ist. Als Konsequenz erhal- ten wir, dass das Verkleben von metrischen R¨aumen entlang extern hyperkon- vexer oder stark konvexer Teilmengen die Hyperkonvexit¨at erhaltet. Wir geben auch eine Klassifizierung fur¨ Verklebungen von hyperkonvexen Vektorr¨aumen entlang eines linearen Unterraumes. In einem zweiten Teil analysieren wir, wann ein lokal hyperkonvexer me- trischer Raum hyperkonvex ist. Zu diesem Zweck beweisen wir einen Cartan- Hadamard-Satz fur¨ metrische R¨aume, in denen wir lokal Geod¨aten mit einer Konvexit¨atseigenschaft w¨ahlen k¨onnen. Es folgt, dass ein vollst¨andiger, ein- fach-zusammenh¨angender, lokal kompakter und lokal hyperkonvexer metrischer Raum mit einer L¨angenmetrik, der lokal endliche kombinatorische Dimension hat, ein hyperkonvexer metrischer Raum ist. Des Weiteren besch¨aftigen wir uns mit schw¨acheren Formen der Hyper- konvexit¨at. Wir zeigen, dass fur¨ n ≥ 3 jeder vollst¨andige, fast n-hyperkonvexe metrische Raum n-hyperkonvex ist, und beweisen dann, dass jeder vollst¨andige, 4-hyperkonvexe metrische Raum bereits n-hyperkonvex fur¨ alle n 2 N ist. Anschliessend untersuchen wir Konvexit¨atseigenschaften von schwach ex- tern hyperkonvexen Teilmengen und verbinden diese mit Lokal-global-Resulta- ten fur¨ diese Mengen. Dies fuhrt¨ unter anderem zu einem Helly-artigen Satz fur¨ n schwach extern hyperkonvexe Teilmengen von l1. Zum Schluss wenden wir uns noch kurz den Medianr¨aumen und Wurfel-¨ komplexen zu. Wir beweisen, dass hyperkonvex metrisierte Wurfelkomplexe¨ mit der entsprechenden Metrik CAT(0)-R¨aume sind. Zudem konstruieren wir eine bi-Lipschitz-¨aquivalente, hyperkonvexe Metrik auf geod¨atischen Medianr¨aumen und zeigen, dass ein lokaler Medianraum mit einem geod¨atischen Bicombing ein Medianraum ist. iii Abstract Hyperconvexity is an intersection property of balls in metric spaces and was introduced in 1956 by N. Aronszajn and P. Panitchpakdi to study extensions of uniformly continuous transformations. In this thesis, we study different properties of hyperconvex metric spaces. First, we investigate gluings of such spaces along isometric subsets. For weakly externally hyperconvex gluing sets, we can give necessary and sufficient condi- tions, so that the resulting space is hyperconvex as well. As a consequence, we get that the gluing of metric spaces along externally hyperconvex or strongly convex subsets preserves hyperconvexity. We then give a classification for glu- ings of hyperconvex vector spaces along linear subspaces. In the second part, we analyze under which conditions a locally hyperconvex metric space is hyperconvex. For this purpose, we prove a Cartan-Hadamard Theorem for metric spaces with a local geodesic bicombing. It follows that a complete, simply-connected, locally compact and locally hyperconvex metric space with finite combinatorial dimension, endowed with the length metric, is hyperconvex. Furthermore, we consider some relaxed notions of hyperconvexity. We show that, for n ≥ 3, every complete, almost n-hyperconvex metric space is n-hyper- convex and prove that every complete, 4-hyperconvex metric space is n-hyper- convex for every n 2 N. Afterwards, we investigate convexity of weakly externally hyperconvex sub- sets and connect them with local-to-global results for these sets. This leads to n a Helly-type theorem for weakly externally hyperconvex subsets of l1. Finally, we turn our attention to median metric spaces and cube complexes. We prove that cube complexes which are hyperconvex with respect to some metric also possess a metric such that they become CAT(0)-spaces. Moreover, we construct a bi-Lipschitz equivalent hyperconvex metric on geodesic median metric spaces and show that a locally median metric space with a geodesic bicombing is a median metric space. v Acknowledgments First of all, I would like to thank Prof. Dr. Urs Lang for his encouragement and guidance during the past years. He was always ready to take time for my questions and could give me the advise needed. I am also indebted to Prof. Dr. Rafael Esp´ınolafor agreeing to act as co-examiner. I joyfully remember my visit to Sevilla. Moreover, I really appreciated the companionship with my colleagues from Assistant Groups 1 & 4 of the Department of Mathematics at ETH Z¨urich. Especially, I would like to mention my office mate Christian, who always had a helping hand. With Nicolas I could also conduct projects beyond mathematics. Ma¨eland Giuliano were excellent coworkers and travel companions. Above all, I thank my parents for their unconditional love and support. I gratefully acknowledge financial support from the Swiss National Science Foundation. vii CONTENTS Contents Abstract v I Introduction 1 II Hyperconvex Metric Spaces 7 II.1 Basic Properties . 7 II.2 An Intersection Property for Externally Hyperconvex Subsets 9 II.3 Weakly Externally Hyperconvex Subsets . 13 II.4 Retracts . 21 III Gluing Hyperconvex Metric Spaces 23 III.1 Gluing along Weakly Externally Hyperconvex Subsets . 23 III.2 Gluing Isometric Copies . 29 III.3 Gluing Hyperconvex Linear Spaces . 31 IV Local to Global 43 IV.1 The Cartan-Hadamard Theorem for Metric Spaces with Local Geodesic Bicombings . 43 IV.2 Locally Hyperconvex Metric Spaces . 49 IV.3 Absolute 1-Lipschitz Neighborhood Retracts . 51 V Finite Hyperconvexity 55 V.1 Basic Properties . 55 V.2 Almost n-Hyperconvex Metric Spaces . 56 V.3 4-Hyperconvex Metric Spaces . 61 VI Convexity of Weakly Externally Hyperconvex Subsets 65 VI.1 Main Results . 65 VI.2 σ-Convexity . 66 VI.3 Locally Weakly Externally Hyperconvex Subsets . 69 VII Hyperconvex Metrics on Median Metric Spaces 75 VII.1 Median Metric Spaces . 75 VII.2 Cube Complexes . 77 VII.3 Hyperconvex Metrics on Median Metric Spaces . 83 VII.4 Locally Median Metric Spaces . 86 ix CONTENTS A Some Results on Geodesic Bicombings 87 A.1 Reversible Geodesic Bicombings . 87 A.2 A Non-Consistent Convex Geodesic Bicombing . 89 Bibliography 97 Index 101 x Chapter I Introduction Hyperconvex metric spaces appear in various contexts. They were introduced by N. Aronszajn and P. Panitchpakdi and later studied by J. Lindenstrauss to receive extension results for linear operators between Banach spaces [AP56, Lin64]. Furthermore, they occur as the injective hull or tight span of a metric space [Isb64, Dre84], which has applications to geometric group theory [Lan13] and theoretical computer science [CL91, CL94]. Hyperconvex metric spaces also play an important role in metric fixed point theory [EK01]. A metric space (X; d) is called hyperconvex if for every family fB(xi; ri)gi2I of closed balls with d(xi; xj) ≤ ri + rj, we have \ B(xi; ri) 6= ;: i2I In other words, (X; d) is a metrically convex metric space whose closed balls have the binary intersection property; compare [BL00, Definition 1.3]. Already N. Aronszajn and P. Panitchpakdi showed that hyperconvex metric spaces are the same as injective metric spaces and absolute 1-Lipschitz retracts [AP56]. From the construction of the injective hull by J. Isbell [Isb64] it follows that hyperconvex metric spaces possess a geodesic bicombing which is invariant un- der isometries [Lan13, Proposition 3.8]. This means that we can select geodesics in such a way that the distance between them fulfills a weak convexity property. Therefore, we can look at hyperconvex metric spaces as some kind of weakly negatively curved spaces. It turns out that many results for CAT(0) and Buse- mann spaces transfer to metric spaces with geodesic bicombings [DL16, Des16] and hence also to hyperconvex metric spaces. In this thesis, we continue in this spirit by providing gluing results for hy- perconvex metric spaces and proving a local-to-global theorem for hypercon- vex metric spaces which is based on a Cartan-Hadamard Theorem for metric spaces with geodesic bicombings. Furthermore, we investigate finite hypercon- vexity and study σ-convexity of weakly externally hyperconvex subsets. Then a section on median metric spaces follows, where we construct a bi-Lipschitz equivalent hyperconvex metric. In the appendix we finally include some further results on geodesic bicombings. 1 CHAPTER I. INTRODUCTION
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