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Research Collection Research Collection Doctoral Thesis The injective hull of hyperbolic groups Author(s): Moezzi, Arvin Publication Date: 2010 Permanent Link: https://doi.org/10.3929/ethz-a-006060216 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. 18860 The Injective Hull of Hyperbolic Groups A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by Arvin Moezzi Dipl. Math. ETH Zurich born 27.08.1976 citizen of Basel BS, Switzerland accepted on the recommendation of Prof. Dr. Urs Lang, examiner Prof. Dr. Viktor Schroeder, co-examiner 2010 For my family Abstract The construction of the injective hull of a metric space goes back to Isbell in the sixties. In this thesis we investigate this construction in the setting of word metrics of finitely generated groups, hyperbolic groups in particular. There is a natural action of the group on its injective hull. We are mainly interested when this action is properly discontinuous and cocompact; we give a necessary condition. Furthermore, we introduce the concept of Axiom Y and show for a hyperbolic group satisfying Axiom Y that this action is cocompact and properly discontinuous. Moreover, in this case the injective hull can be given explicitly the structure of a polyhedral complex such that each cell in itself is injective and the group acts by cellular isometries on it. We give some simple examples of groups satisfying Axiom Y and a counter example of a Cayley graph of a hyperbolic group that does not satisfy Axiom Y. Zusammenfassung Die injektive H¨ulle von metrischen R¨aumen geht zur¨uck auf Isbell in den Sechzigern. Wir betrachten in dieser Arbeit die injektive H¨ulle einer endlich erzeugten Gruppe versehen mit der Wort Metrik - hyperbolische Grup- pen aber im besonderen. Die Gruppe selbst operiert auf nat¨urliche Weise isometrisch auf ihrer injektiven H¨ulle. Es stellt sich die Frage, wann diese Gruppenaktion eigentlich und cocompakt ist. Wir geben eine notwendige Bedingung an die Gruppe. Desweiteren f¨uhren wir die Eigenschaft Ax- iom Y ein und zeigen, dass jede hyperbolische Gruppe, die das Axiom Y erf¨ullt, eigentlich und cocompakt auf ihrer injektiven H¨ulle operiert. In diesem Fall l¨asst sich weiter zeigen, dass die injektive H¨ulle die Struktur eines polyedrischen Komplexes tr¨agt, wobei jede Zelle f¨ur sich wieder injek- tiv ist und die Gruppe zellul¨ar operiert. Wir geben einige einfache Beispiele von Gruppen, die das Axiom Y erf¨ullen, und auch ein Gegenbeispiel eines Cayley Graphen einer hyperbolischen Gruppe, welcher Axiom Y nicht erf¨ullt. Danksagung Die vorliegende Doktorarbeit w¨are ohne die Hilfe und Unterst¨utzung an- derer in dieser Form nicht m¨oglich gewesen. Daher m¨ochte ich gerne fol- genden Menschen an dieser Stelle danken; allen voran meinem Doktorvater, Professor Urs Lang. Er war es, der mich in die metrische Geometrie und geometrische Gruppentheorie einf¨uhrte. Dieses faszienerende Teilgebiet der Mathematik w¨are mir ansonsten wohl verschlossen geblieben. Auch nahm er sich stets die n¨otige Zeit f¨ur Fragen und hat die vorliegende Arbeit mit wichti- gen Ideen und Konzepten vorangetrieben; ohne seine grosse Mithilfe w¨are die Doktorarbeit niemals zu dem geworden, was sie ist. Viel Zeit investierte er zudem, die Arbeit mit der von ihm gewohnten Genauigkeit durchzuar- beiten und half mir so, auch die letzten Fehler auszumerzen. Vorallem aber m¨ochte ich ihm daf¨ur danken, dass er mir vertraute und an meine F¨ahigkeiten glaubte - tausend Dank daf¨ur! Ein grosses Dankesch¨on gilt auch dem Koreferenten, Professor Viktor Schroeder, der sich die M¨uhe machte, meine Arbeit zu lesen und f¨ur die m¨undliche Doktorpr¨ufung kurzfristig nach Z¨urich zu reisen. Mein besonderer Dank gilt auch zwei guten Freunden von mir, Johnny Micic und Driton Komani. Beide haben grosse Teile der Arbeit mehrfach korrekturgelesen und haben mich bei der Pr¨ufungsvorbereitung tatkr¨aftig unterst¨utzt. Sehr wichtig f¨ur mich waren auch die zahlreichen mathematis- chen Debatten mit ihnen zu allen m¨oglichen Themen. Diese Diskussionen f¨uhrten mir immer wieder vor Augen was ich am liebsten mache: Mathe- matik. Der lange Weg zum Abschluss der Doktorarbeit w¨are sicherlich unendlich l¨anger gewesen ohne die Freundschaft und die moralische Unterst¨utzung vieler. Nennen m¨ochte ich an dieser Stelle Fabian Roth, Yves Hauser, B´en´edicte Gros, Thilo Schlichenmaier, Stefan Wenger, Michael Anderegg, Driton Komani, Johnny Micic, Leandra Simitovic, Theo Buehler, Beat Steiner, Kathrin Signer, Luzia Huggentobler, Demian Wismer, Kascha, Robin Krom, Ivo K¨ahlin, Lorenz Reichel, Thomas Huber, Ralph Kirchhofer und Jan Oppliger. Vielen Dank, dass ihr f¨ur mich da wart. Und vielen Dank f¨ur eine sehr gelungene Doktorfeier. Danken m¨ochte ich auch Tim Boin f¨urs Korrekturlesen der deutschen Passagen. Zuletzt und ganz besonders m¨ochte ich meiner Familie danken. Ohne ihre Liebe und Unterst¨utzung, die man leider allzu oft als selbstverst¨andlich betrachtet, h¨atte ich dieses Ziel niemals erreicht. Danke f¨ur eure St¨utze und eine Konstante in meinem Leben! Contents 1 Isbell’s Injective Hull 5 1.1 Preliminaries . 5 1.2 isoM-Injectivity ........................ 8 1.3 Bicombing and isoperimetric inequality . 13 1.4 Isbell’s injective hull . 15 1.5 Gromov–Hausdorffconvergence . 21 1.6 Compactmetricspaces . .. .. 26 1.7 Stronglyconvexsubspaces . 30 1.8 Normedvectorspaces. .. .. 35 n 1.8.1 Affine, injective subspaces of R∞ ........... 41 n 1.8.2 Injective polyhedrons in R∞ .............. 46 1.9 Finitemetricspaces....................... 51 2 Groups and their Injective Hull 57 2.1 Cayleygraphs.......................... 59 2.2 Conetypes............................ 62 2.3 Combablegroups ........................ 63 2.4 Hyperbolicgroups........................ 69 2.5 AxiomY............................. 72 2.6 Injectivehullofhyperbolicgroups . 73 3 Examples and a counter-example 83 3.1 Hyperbolicgroups........................ 83 3.1.1 ACounterexample ................... 83 3.1.2 Small cancellation groups . 86 3.2 Abeliangroups ......................... 90 A The Stone–Cechˇ compactification 95 B Group actions on geodesic metric spaces 99 CTheGrowthofaDehnFunction 101 Introduction A metric space I is said to be isoM-injective if every 1-Lipschitz map, f : A → I, can be extended to B whenever A is isometrically embedded in B. In other words, there exists a 1-Lipschitz map f¯: B → I such that the diagram f A / I ? ¯ f B commutes. isoM-injective metric spaces share nice metric properties; they are contractible and complete geodesic metric spaces, and they all admit an isoperimetric inequality of euclidean type in the class of integral metric currents. One may wonder whether every X can be isometrically embedded into some isoM-injective metric space I and if there is a smallest such space, in the sense that whenever X can be isometrically embedded into some other isoM-injective metric space J, there exists an isometric embedding of I into J such that X / Io J commutes. In this case I is unique up to an isometry, and it is said to be the injective hull of X; we shall write EX for I. In [22] Isbell shows that every metric space X can indeed be embedded isometrically into a smallest isoM-injective metric space EX. He gives an explicit description of EX in terms of some function space on X. We will show in Theorem 1.55 that the assignment X → EX is continuous with respect to the Gromov–Hausdorff distance; that is E E Xn →dGH X ⇒ Xn →dGH X. Furthermore, in Theorem 2.58 we are going to show that Gromov hyperbol- icity for geodesic metric spaces is preserved under this assignment: 4 Theorem. The injective hull of a geodesic hyperbolic metric space is hy- perbolic. However, in this thesis we are mostly interested in the following ques- tion: when does a finitely generated group (Γ,S) act nicely, i.e. properly discontinuous and cocompactly, on EΓS, where ΓS denotes the metric space Γ with the word metric; in particular what is the metric structure of EΓS in this case? We shall give in 2.39 a necessary condition for Γ to act nicely on EΓS: Proposition. Every group that acts properly discontinuous and cocom- pactly on an injective metric space is a combable group. But this condition is by no means sufficient. The finitely generated abelian group Zn, for example, with the standard generating set is a combable group but does not act nicely on EZn; compare page 68. In Section 2.5 we introduce the concept of Axiom Y(δ), a metric property which postulates that for any three points x, z, z′ with d(z, z′) ≤ δ one can find a point y such that d(z,y) ≤ R = R(δ) and d(x, z) = d(x, y)+ d(y, z) d(x, z′) = d(x, y)+ d(y, z′). The main result of this thesis, Theorem 2.71, can then be formulated as follows. Theorem. Suppose (Γ,S) is a δ-hyperbolic group and ΓS satisfies Axiom Y(δ). Then, EΓS is a proper locally finite polyhedral complex with finitely many isometry types of cells, and Γ acts properly discontinuous and cocom- pactly on EΓS by cellular isometries. The cell structure on EΓS can be given explicitly in terms of some admis- sible graphs on ΓS, and every cell endowed with the induced metric is itself isoM-injective; see Theorem 2.70. Moreover, we shall see in 2.72 that every tangent cone of EΓS is an isoM-injective metric space. For further study, we hope that this property would give some good link conditions on the cell structure of EΓS, allowing us to alter the metric on the cells in such a way that EΓS becomes a CAT(0) space.
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