\rtp-impa" i i 2005/5/12 page 1 i i Dynamics of infinite-dimensional groups and Ramsey-type phenomena Vladimir Pestov May 12, 2005 i i i i \rtp-impa" i i 2005/5/12 page 2 i i i i i i \rtp-impa" i i 2005/5/12 page i i i Contents Preface iii 0 Introduction 1 1 The Ramsey{Dvoretzky{Milman phenomenon 17 1.1 Finite oscillation stability . 17 1.2 First example: the sphere S1 . 27 1.2.1 Finite oscillation stability of S1 . 27 1.2.2 Concentration of measure on spheres . 32 1.3 Second example: finite Ramsey theorem . 41 1.4 Counter-example: ordered pairs . 47 2 The fixed point on compacta property 49 2.1 Extremely amenable groups . 49 2.2 Three main examples . 55 2.2.1 Example: the unitary group . 55 2.2.2 Example: the group Aut (Q; ) . 60 ≤ 2.2.3 Counter-example: the infinite symmetric group 62 2.3 Equivariant compactification . 63 2.4 Essential sets and the concentration property . 66 2.5 Veech theorem . 71 2.6 Big sets . 76 2.7 Invariant means . 81 3 L´evy groups 95 3.1 Unitary group with the strong topology . 95 i i i i i \rtp-impa" i i 2005/5/12 page ii i i ii CONTENTS 3.2 Group of measurable maps . 98 3.2.1 Construction and example . 98 3.2.2 Martingales . 101 3.2.3 Unitary groups of operator algebras . 110 3.3 Groups of transformations of measure spaces . 115 3.3.1 Maurey's theorem . 115 3.3.2 Group of measure-preserving transformations . 117 3.3.3 Full groups . 123 3.4 Group of isometries of the Urysohn space . 133 3.4.1 Urysohn universal metric space . 133 3.4.2 Isometry group . 143 3.4.3 Approximation by finite groups . 146 3.5 L´evy groups and spatial actions . 155 4 Minimal flows 165 4.1 Universal minimal flow . 165 4.2 Group of circle homeomorphisms . 168 4.2.1 Example . 168 4.2.2 Large extremely amenable subgroups . 169 4.3 Infinite symmetric group . 173 4.4 Maximal chains and Uspenskij's theorem . 176 4.5 Automorphism groups of Fra¨ıss´e structures . 180 4.5.1 Fra¨ıss´e theory . 180 4.5.2 Extremely amenable subgroups of S1 . 185 4.5.3 Minimal flows of automorphism groups . 191 5 Emerging developments 199 5.1 Oscillation stability and Hjorth theorem . 199 5.2 Open questions . 213 Bibliography 223 Index 239 i i i i \rtp-impa" i i 2005/5/12 page iii i i Preface This set of lecture notes is an attempt to present an account of basic ideas of a small new theory located on the crossroads of geometric functional analysis, topological groups of transformations, and combinatorics. The presentation is certainly incomplete. The lecture notes are organized on a \need-to-know" basis, and in particular I try to resist the temptation to achieve extra generality for its own sake, or else introduce concepts or results that will not be used more or less immediately. The notes have grown out of mini-courses given by me at the Nipissing University, North Bay, Canada (26{27 May 2004), at the 8i`eme at´elier sur la th´eorie des ensembles, CIRM, Luminy, France (20{24 september, 2004), and in preparation for a mini-course to be given at IMPA, Rio de Janeiro, Brazil (25{29 July 2005). Acknowledgements In this area of study, I was most influenced by conversations with Eli Glasner, Misha Gromov, and Vitali Milman, to whom I am pro- foundly grateful. Special thanks go to my co-authors Thierry Giordano, Alekos Kechris, Misha Megrelishvili, Stevo Todorcevic, and Volodja Uspen- skij, from whom I have learned much, and I am equally grateful to Pierre de la Harpe and A.M. Vershik. I want to acknowledge important remarks made by, and stimulat- ing discussions with, Joe Auslander, V. Bergelson, Peter Cameron, Michael Cowling, Ed Granirer, John Clemens, Ilijas Farah, David Fremlin, Su Gao, C. Ward Henson, Greg Hjorth, Menachem Koj- iii i i i i \rtp-impa" i i 2005/5/12 page iv i i iv PREFACE man, Julien Melleray, Ben Miller, Carlos Gustavo \Gugu" T. de A. Moreira, Christian Rosendal, Matatyahu Rubin, Slawek Solecki, and Benjy Weiss. This manuscript has much benefitted from stimulating comments and suggestions made by V.V. Uspenskij, too numerous for most of them to be acknowledged. Many useful remarks on the first version of the notes were also made by T. Giordano, E. Glasner, X. Dom´ınguez, I. Farah, A. Kechris, J. Melleray, V.D. Milman, C. Rosendal, and A.M. Vershik. As always, the author bears full responsibility for the remaining errors and omissions, which are certainly there; the hope is that none of them is too serious. The reader is warmly welcomed to send me comments at <[email protected]>. During the past nine years that I have been working, on and off, in this area, the research support was provided by: two Marsden Fund grants of the Royal Society of New Zealand (1998{2001, 2001{03), Victoria University of Wellington research development grant (1999), University of Ottawa internal grants (2002{04, 2004{ ), and NSERC discovery grant (2003{ ). The Fields institute has supported the workshop \Interactions between concentration phenomenon, trans- formation groups, and Ramsey theory" (University of Ottawa, 20-22 October 2003). Vladimir Pestov Gloucester, Ontario, 12 May 2005 i i i i \rtp-impa" i i 2005/5/12 page 1 i i Chapter 0 Introduction The “infinite-dimensional groups" in the title of this set of lecture notes refer to a vast collection of concrete groups of automorphisms of various mathematical objects. Examples include unitary groups of Hilbert spaces and, more generally, operator algebras, the infi- nite symmetric group, groups of homeomorphisms, groups of au- tomorphisms of measure spaces, isometry groups of various metric spaces, etc. Typically such groups are supporting additional struc- tures, such as a natural topology, or sometimes the structure of an infinite-dimensional Lie group. In fact, it appears that those ad- ditional structures are often encoded in the algebraic structure of the groups in question. The full extent of this phenomenon is still unknown, but, for instance, a recent astonishing result by Kechris and Rosendal [100] states that there is only one non-trivial separable group topology on the infinite symmetric group S1. It could also well be that there are relevant structures on those groups that have not yet been formulated explicitely. Those infinite-dimensional groups are being studied from a num- ber of different viewpoints in various parts of mathematics: most no- tably, representation theory [135, 128], logic and descriptive set the- ory [11, 91], infinite-dimensional Lie theory [85, 136], theory of topo- logical groups of transformations [5, 186] and abstract harmonic anal- ysis [17, 37], as well as from a purely group theoretic viewpoint [9, 18] and from that of set-theoretic topology [164]. Together, those studies 1 i i i i \rtp-impa" i i 2005/5/12 page 2 i i 2 [CAP. 0: INTRODUCTION mark a new chapter in the theory of groups of transformations, a post Gleason{Yamabe{Montgomery{Zippin development [126]. The present set of lecture notes outlines an approach to the study of infinite-dimensional groups based on the ideas originating in geo- metric functional analysis, and exploring an interplay between the dynamical properties of those groups, combinatorial Ramsey-type theorems, and geometry of high-dimensional structures (asymptotic geometric analysis). The theory in question is a result of several developments, some of which have happened independently of, and in parallel to, each other, so it is a \partially ordered set." On the other hand, each time those developments have been absorbed into the theory and put into connection to each other, so the set is, hopefully, \directed." It is not clear how far back in time one needs to go towards the beginnings. For example, one of the three components of the theory | the phenomenon of concentration of measure, the subject of study of the modern-day asymptotic geometric analysis | was explicitely described and studied in the 1922 book by Paul L´evy [103] for spheres. On the other hand, another major manifestation of the phenomenon { the Law of Large Numbers { was stated by Emile´ Borel in the modern form back in 1904, see a discussion in [120]. And according to Gromov [76], the concentration of measure was first extensively used by Maxwell in his studies of statistical physics (Maxwell{Boltzmann distribution law, 1866). We will take as the point of departure for interactions between geometry of high-dimensional structures, topological transformation groups, and combinatoric the Dvoretzky theorem [38, 39], regarded in light of later results by Vitali Milman. As Aryeh Dvoretzky remarks in his 1959 Proc. Nat. Acad. Sci. USA note [38], \this result, or various weaker variations, have often been conjectured. However, the only explicit statement of this conjecture in print known to the author is in a recent paper by Alexandre Grothendieck" [80]. Interestingly, Grothendieck's paper was published (in the Bol. Soc. Mat. S~ao Paulo) during the period when he was a visiting professor at the Universidade de S~ao Paulo, Brazil (1953{54). i i i i \rtp-impa" i i 2005/5/12 page 3 i i 3 Theorem 0.0.1 (Dvoretzky theorem, 1959). For every " > 0, each n-dimensional normed space X contains a subspace E of dimen- sion dim E = k c"2 log n, which is Euclidean to within ": ≥ d (E; `2(k)) 1 + ": BM ≤ Here dBM is the multiplicative Banach{Mazur distance between two isomorphic normed spaces: d (E; F ) = inf T T −1 : T is an isomorphism : BM fk k · g (In geometric functional analysis, isomorphic normed space are those isomorphic as topological vector spaces, that is, topologically isomor- phic.) The geometric meaning of the Banach-Mazur distance is clear from the following figure.