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Ramsey Properties of Finite Measure Algebras And Contemporary Mathematics Ramsey properties of finite measure algebras and topological dynamics of the group of measure preserving automorphisms: some results and an open problem A.S. Kechris, M. Soki´c,and S. Todorcevic Dedicated to Hugh Woodin on his 60th birthday Abstract. We study in this paper ordered finite measure algebras from the point of view of Fra¨ıss´eand Ramsey theory. We also propose an open problem, which is a homogeneous version of the Dual Ramsey Theorem of Graham- Rothschild, and derive consequences of a positive answer to the study of the topological dynamics of the automorphism group of a standard probability space and also the group of measure preserving homeomorphisms of the Cantor space. 1. Introduction In this paper we continue the theme developed in the paper Kechris-Pestov- Todorcevic [KPT] of exploring the connections between structural Ramsey the- ory and topological dynamics of Polish groups, especially the study of extreme amenability and the calculation of universal minimal flows. A topological group G is called extremely amenable (or said to have the fixed point on compacta property) if every continuous action of G on a (non-empty, Haus- dorff) compact space has a fixed point. There is now a plethora of Polish groups known to have this very strong fixed point property, e.g., the unitary group of a separable infinite-dimensional Hilbert space (Gromov-Milman [GM]), the auto- morphism group of the ordered rationals (Pestov [P1]), the isometry group of the Urysohn space (Pestov [P2]), the automorphism group of a standard measure space (Giordano-Pestov [GP]), etc. An excellent exposition of the theory of such groups and its connections with other areas, such as asymptotic geometric analysis, in par- ticular concentration of measure phenomena, and Ramsey theory, can be found in 2010 Mathematics Subject Classification. Primary 03C13, 03C15, 05D10, 37B05, 37A15; Secondary 54H20. A.S.K. was partially supported by NSF grants DMS-0455285, DMS-0968710 and DMS- 1464475. S.T. was partially supported by grants from NSERC and CNRS. c 2009 American Mathematical Society 1 2 A.S. KECHRIS, M. SOKIC,´ AND S. TODORCEVIC the recent book Pestov [P3]. As pointed out in this monograph there are primar- ily two main known general techniques for establishing extreme amenability: one analytical, via the concept of a L´evygroup and concentration of measure, and the other combinatorial, via finite Ramsey theory. In [KPT] structural Ramsey theory has been used to prove extreme amenabil- ity of many automorphism groups of countable structures, where concentration of measure techniques are not directly applicable. It is interesting to investigate to what extent Ramsey methods can be used to establish extreme amenability for other types of groups of a more analytic or topological nature and a first step in that direction was taken in [KPT], where a new proof of the extreme amenabil- ity of the isometry group of the Urysohn space, Iso(U), was established, a result originally proved in Pestov [P2] using concentration of measure (see also Pestov [P3]). An important dividend of this approach is that it motivates the discovery of new classes that satisfy the Ramsey Property. For example, the Ramsey theory approach to the extreme amenability of Iso(U) led [KPT] to conjecture that the class of finite metric spaces has the Ramsey Property, which was indeed established in Ne´setˇril[N]. Until now this example is the only one in which combinatorial Ramsey tech- niques are used to replace analytical arguments in establishing extreme amenability. We propose in this paper to apply such techniques to another important example of an extremely amenable group, namely the automorphism group Aut(X; µ) of a standard probability measure space (X; µ). This result was originally established in Giordano-Pestov [GP] by concentration of measure arguments. Again our ap- proach leads us to propose a new Ramsey theorem, a homogeneous version of the Dual Ramsey Theorem. A proof of this Ramsey theorem would also allow us to compute the universal minimal flow of another interesting group, namely the group of measure preserving homeomorphisms of the Cantor space (with the topology it inherits as a closed subgroup of the homeomorphism group). We now proceed to describe more precisely the contents of this paper. First we recall some basic notions that are discussed in detail in Section 2. A class of finite structures in a given countable language is said to be a Fra¨ıss´e class if it is unbounded (i.e., contains structures of arbitrarily large finite cardinal- ity), has only countably many members, up to isomorphism, is hereditary (under embeddability) and satisfies the joint embedding and amalgamation properties. For such a class K there is a unique countably infinite structure, its Fra¨ıss´elimit, Flim(K), which is locally finite (finitely generated substructures are finite), ultraho- mogeneous (isomorphisms between finite substructures extend to automorphisms) and whose class of finite substructures, up to isomorphism, coincides with K. In this paper we study K = OMBAQ2 , the class of naturally ordered finite mea- sure algebras with measure taking values in the dyadic rationals. These are struc- tures of the form A = hA; ^; _; 0; 1; µ, <i, where hA; ^; _; 0; 1i is a finite Boolean algebra, µ : A ! [0; 1] a positive measure on A with values in the set Q2 of dyadic rationals, and < an ordering induced antilexicographically by an ordering of the atoms. We first show the following result. Theorem 1.1. The class K = OMBAQ2 is a Fra¨ıss´eclass. It is not hard then to compute that the Fra¨ıss´elimit of this class is hB1; λ, ≺i; RAMSEY PROPERTIES OF FINITE MEASURE ALGEBRAS 3 where B1 is the algebra of clopen subsets of the Cantor space 2N (i.e., the countable atomless Boolean algebra), λ is the usual product measure on 2N and ≺ the so-called canonical ordering on B1, defined by ∼ hB1; ≺i = Flim(OBA); where OBA is the class of naturally ordered finite Boolean algebras (see [KPT], 6,(D)). Next recall that a class K has the Ramsey Property (RP) if for any two struc- tures A ≤ B in K (where A ≤ B means that A can be embedded in B) and N = 1; 2;::: , there is a C 2 K, with B ≤ C, such that for any coloring with N colors of the set of isomorphic copies of A in C, there is a copy B0 of B in C so that all the copies of A contained in B0 have the same color. We now propose the following problem: Problem 1.2. Is it true that the class K = OMBAQ2 has the Ramsey Prop- erty? In fact this statement can be viewed as a homogeneous version of the Dual Ramsey Theorem of Graham-Rothschild [GR] which asserts the following: Given k < ` and N there is m > ` so that for any coloring with N colors of the set of equivalence relations on f1; : : : ; mg with k many classes, there is an equivalence relation F with ` many classes, such that any coarser than F equivalence relation with k many classes has the same color. It is easy to show that the Dual Ramsey Theorem is equivalent to the assertion that the class of finite Boolean algebras BA has the Ramsey Property. One can now see that the class K = OMBAQ2 has the Ramsey Property iff the class of homogeneous finite measure Boolean algebras with measure taking values in Q2 has the Ramsey Property. Here homogeneous means that all atoms have the same measure. This in turn translates to the following homogeneous version of the Dual Ramsey Theorem. Call an equivalence relation homogeneous if all its classes have the same cardinality. Then we have that for each k < `; N, there is m > ` such that for any coloring with N colors of the set of homogeneous equivalence relations on f1;:::; 2mg with 2k many classes, there is a homogeneous equivalence relation F on f1; 2;:::; 2mg with 2` many classes such that all coarser homogeneous equivalence relations with 2k many classes have the same color. In fact there is no reason to restrict ourselves to numbers that are powers of two, except for some obvious restrictions: Let below k ` mean that k < ` and k divides `. Then we have the following problem. Problem 1.3 (Homogeneous Dual Ramsey). Is it true that given N and k `, there is m ` such that for any coloring with N colors of the homogeneous equivalence relations on f1; : : : ; mg with k many classes, there is a homogeneous equivalence relation F with ` many classes, so that all homogeneous equivalence relations coarser than F with k many classes have the same color? Assuming now a positive answer to Problem 1.2, we can apply the general theory of [KPT] to give a combinatorial proof of the extreme amenability for the automorphism group Aut(X; µ) of a standard measure space (X; µ). Clearly we can take X = 2N; µ = λ. By Theorem 1.1, a positive answer to Problem 1.2 and the results in [KPT], Aut(B1; λ, ≺) is extremely amenable. Now (the Polish group) Aut(B1; λ, ≺), with the pointwise convergence topology, can be 4 A.S. KECHRIS, M. SOKIC,´ AND S. TODORCEVIC continuously embedded in a natural way in Aut(2N; λ), where, as usual, Aut(2N; λ) has the standard (weak) topology. Theorem 1.4. The image of the natural embedding of Aut(B1; λ, ≺) in the group Aut(2N; λ) is dense in Aut(2N; λ). It is clear that if an extremely amenable group G embeds densely in a group H, then H is also extremely amenable. Thus, assuming a positive answer to Problem 2, one obtains a new, combinatorial proof of the Giordano-Pestov [GP] result that the automorphism group of a standard measure space, Aut(X; µ), is extremely amenable.
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