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Spaces with Convex Geodesic Bicombings Research Collection Doctoral Thesis Spaces with convex geodesic bicombings Author(s): Descombes, Dominic Publication Date: 2015 Permanent Link: https://doi.org/10.3929/ethz-a-010584573 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. 23109 Spaces with Convex Geodesic Bicombings A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich presented by Dominic Descombes Master of Science ETH in Mathematics citizen of Lignières, NE and citizen of Italy accepted on the recommendation of Prof. Dr. Urs Lang, examiner Prof. Dr. Alexander Lytchak, co-examiner 2015 Life; full of loneliness, and misery, and suffering, and unhappiness — and it's all over much too quickly. – Woody Allen Abstract In the geometry of CAT(0) or Busemann spaces every pair of geodesics, call them α and β, have convex distance; meaning d ◦ (α, β) is a convex function I → R provided the geodesics are parametrized proportional to arc length on the same interval I ⊂ R. Therefore, geodesics ought to be unique and thus even many normed spaces do not belong to these classes. We investigate spaces with non-unique geodesics where there exists a suitable selection of geodesics exposing the said (or a similar) convexity property; this structure will be called a bicombing. A rich class of such spaces arises naturally through the construction of the injective hull for arbitrary metric spaces or more generally as 1- Lipschitz retracts of normed spaces. In particular, the injective hulls of word hyperbolic groups admit bicombings. We are able to generalize a variety of classical results to this broader context of weak non-positive curvature including a flat torus theorem and an asymptotic rank theorem (an equivalence of several notions of embeddings of flats) — results first shown for manifolds of non-positive curvature. New tools are needed in this less restricted setting. We employ a well-known, elegant barycenter construction that will be the key to many proofs. v Zusammenfassung In CAT(0)- oder Busemann-Räumen gilt für beliebige zwei Geodäten, nennen wir diese α und β, dass der Abstand d ◦ (α, β) eine konvexe Funktion I → R ist. Vorausgesetzt wird dabei nur, dass die beiden Geodäten auf demselben Intervall I definiert und pro- portional zur Bogenlänge parametrisiert sind. Geodäten sind daher eindeutig, was für viele interessante Räume eine zu restriktive Forderung darstellt. Wir untersuchen da- her Räume, in welchen wir eine Auswahl an Geodäten treffen können, so dass die obige Konvexitätsbedingung (oder eine ähnliche) zumindest für diese Teilmenge gilt; diese Struktur nennen wir ein bicombing. Diese Klasse umfasst nun auch alle normierten Räume und allgemeiner alle 1-Lipschitz-Retrakte solcher. Weitere Beispiele von Räumen mit bicombings entstehen auf natürliche Weise durch die Konstruktion injektiver Hüllen metrischer Räume und im Speziellen besitzen injektive Hüllen hyperbolischer Gruppen diese Struktur. Wir beweisen zentrale klassische Resultate in diesem allgemeineren Rah- men schwacher, nichtpositiver Krümmung. Darunter einen Satz über flache Tori und ein Theorem über den asymptotischen Rang (eine Äquivalenz über die Einbettbarkeit flacher Räume) — Resultate welche ursprünglich für Mannigfaltigkeiten nichtpositiver Krümmung gezeigt wurden. Manchmal führen die klassischen Beweispfade zum Ziel, häufig aber auch nicht. Unter den neuen Hilfsmitteln, die nun benötigt werden, befindet sich auch eine Schwerpunktkonstruktion, von welcher wir wiederholt und an kritischen Stellen Gebrauch machen. vii Acknowledgments Most of all, I want to express my gratitude to my advisor Urs Lang. Without his patience, gentle guidance, and firm support through these years, I would not have made it. I could always stop by his office to discuss mathematics and always left it with new ideas and renewed motivation for doing my research. I also thank the Swiss National Science Foundation for supporting me financially. And of course, I am also indebted to Alexander Lytchak for agreeing to act as co-examiner. The time I spent at the department of mathematics would not have been so rewarding without the wonderful people of group 1 & 4. Thank you all! And in particular, I am grateful for the time and fun I had with my office mates Dante Bonolis, Maël Pavón and Michael Th. Rassias. Finally, I thank my family for the support during all of my studies and most of all my girlfriend Julia. ix Contents 1 Introduction .................................... 1 2 Basic concepts ................................... 6 2.1 Metric spaces . 6 2.2 Bicombings . 9 2.3 Group actions . 11 3 Improving bicombings .............................. 12 3.1 From conical to convex bicombings . 12 3.2 Straight curves . 15 3.3 Combinatorial dimension . 18 4 Boundary at infinity ............................... 22 5 Flat planes ..................................... 28 5.1 Flat strips and half-planes . 28 5.2 The Flat Plane Theorem . 32 6 Barycenters ..................................... 36 6.1 The construction . 36 6.2 Linear barycenters . 42 6.3 Integration . 44 7 Semi-simple isometries .............................. 45 7.1 Minimal displacement and axes . 45 7.2 σ-Axes . 48 8 Flat tori ....................................... 52 9 Asymptotic rank .................................. 57 A Injective metric spaces .............................. 64 A.1 Basic definitions and injective hulls . 64 n A.2 A characterization of injective subsets in `∞ . 68 Bibliography ...................................... 74 xi Chapter 1 Introduction The geometry of spaces of global non-positive curvature is largely dominated by the convexity of the distance function. Thus a considerable part of the theory of CAT(0) spaces [Bal, BriH] carries over to Busemann spaces [Bus, Pap] (defined by the property that d ◦ (α, β) is convex for any pair of constant speed geodesics α, β parametrized on the same interval). However, this larger class of spaces has the defect of not being preserved under limit processes. For example, among normed real vector spaces, exactly those with strictly convex norm satisfy the Busemann property, and a sequence of such n norms on R , say, may converge to a non-strictly convex norm. This motivates the study of an even weaker notion of non-positive curvature that dispenses with the uniqueness of geodesics but retains the convexity condition for a suitable selection of geodesics (compare Section 10 in [Kle]). In any normed space, the affine segments t 7→ (1−t)x+ty (t ∈ [0, 1]) provide such a choice. In particular, the relaxed condition carries the potential for simultaneous generalizations of results for non-positively curved and Banach spaces. A selection of geodesics for some metric space (X, d) we call a geodesic bicombing σ and thereby mean a map σ : X × X × [0, 1] → X that singles out a constant speed geodesic σxy := σ(x, y, ·) from x to y (that is, σxy(0) = 0 0 0 x, σxy(1) = y, and d(σxy(t), σxy(t )) = |t − t |d(x, y) for all t, t ∈ [0, 1]) for every pair (x, y) ∈ X × X. Because we work exclusively with bicombings that are composed of geodesics, we suppress the term “geodesic” throughout; a bicombing is thus implicitly understood to be a geodesic bicombing. We are interested in bicombings satisfying one of the following two conditions, each of which implies that σ is continuous and, hence, X is contractible via X ×[0, 1] → X, (x, t) 7→ σox(1−t) for an arbitrary choice of basepoint o ∈ X. We call σ convex if the function t 7→ d(σxy(t), σx0y0 (t)) is convex on [0, 1] (1.1) for all x, y, x0, y0 ∈ X, and we say that σ is conical if 0 0 d(σxy(t), σx0y0 (t)) ≤ (1 − t) d(x, x ) + t d(y, y ) for all t ∈ [0, 1] (1.2) 2 CHAPTER 1. INTRODUCTION and x, y, x0, y0 ∈ X. Obviously every convex bicombing is conical, but the reverse implica- tion does not hold in general (see Example 3.2). For this, in order to pass condition (1.2) to subsegments, one would need to know in addition that σ is consistent in the sense that σpq(λ) = σxy((1 − λ)s + λt) (1.3) whenever x, y ∈ X, 0 ≤ s ≤ t ≤ 1, p := σxy(s), q := σxy(t), and λ ∈ [0, 1]. Thus, every conical and consistent bicombing is convex. Since we always assume our bicombings to be conical at least (for requiring a space to admit a bicombing is nothing more than it being geodesic), we adopt the convention that bicombing shall mean conical (and geodesic) bicombing from now on. Consequently, a consistent bicombing is automatically convex. Furthermore, we will say that a bicombing σ is reversible if σxy(t) = σyx(1 − t) for all t ∈ [0, 1] (1.4) and x, y ∈ X. Consistent and reversible bicombings have also been employed in [HitL, FoeL]. We arrive at the following hierarchy of global non-positive (but non-coarse) curvature conditions (A) ⇒ (B) ⇒ (C) ⇒ (D) ⇒ (E) for a geodesic metric space X: (A) X is a CAT(0) space, (B) X is a Busemann space, (C) X admits a consistent bicombing, (D) X admits a convex bicombing, (E) X admits a (conical) bicombing. Clearly, if X is uniquely geodesic, then (E) ⇒ (B). When X is a normed real vec- tor space, (A) holds if and only if the norm is induced by an inner product (compare Proposition II.1.14 in [BriH]), (B) holds if and only if the norm is strictly convex (see Proposition 8.1.6 in [Pap]), and (C) is always satisfied. In the general case, (C) is stable under limit operations, whereas (B) is not (see, for instance, Section 10 in [Kle] again). It is unclear whether (E) ⇒ (D) and (D) ⇒ (C) without further conditions. The goal of Chapter 3 is to establish these implications under suitable assumptions and to address questions of uniqueness.
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