Introduction(A) We study in this paper some connections between the Fraïssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X. These are usually referred to as (compact) G-flows. Of particular interest is the study of minimal Gflows, those for which every orbit is dense. Every G-flow contains a minimal subflow. A general result of topological dynamics asserts that every topological group G has a universal minimal flow M(G), a minimal G-flow which can be homomorphically mapped onto any other minimal G-flow. Moreover, this is uniquely determined, by this property, up to isomorphism. (As usual a homomorphism π : X → Y between G-flows is a continuous G-map and an isomorphism is a bijective homomorphism.) For separable, metrizable groups G, which are the ones that we are interested in here, the universal minimal flow of G is an inverse limit of manageable, i.e., metrizable G-flows, but itself may be very complicated, for example non-metrizable. In fact, for the "simplest" infinite G, i.e., the countable discrete ones, M(G) is a very complicated compact G-invariant subset of the space βG of ultrafilters on G and is always non-metrizable.Rather remarkably, it turned out that there are non-trivial topological groups G for which M(G) is actually trivial, i.e., a singleton. This is equivalent to saying that G has a very strong fixed point property, namely every G-flow has a fixed point (i.e., a point x such that g • x = x, ∀g ∈ G). (For separable, metrizable groups this is also equivalent to the fixed point property restricted to metrizable G-flows.) Such groups are said to have the fixed point on compacta property or be extremely amenable. The latter name comes from one of the standard characterizations of second countable locally compact amenable groups. A second countable locally compact group G is amenable iff every metrizable G-flow has an invariant (Borel probability) measure. However, no non-trivial locally compact group can be extremely amenable, because, by a theorem of Veech [83], every such group admits a free G-flow (i.e., a flow for which g • x = x ⇒ g = 1 G ). Nontriviality of the universal minimal flow for locally compact groups also follows from the earlier results of . This probably explains the rather late emergence of extreme amenability. Note that the corresponding property
In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many ω‐stable, ℵ0‐categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S∞ has a unique non‐trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).
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