Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups

Abstract

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

A Ramsey theorem for partial orders with linear extensions

“…In recent years, there has been a renewed interest in Structural Ramsey Theory sparked by the discovery in [4] of connections between this area and Topological Dynamics. Paper [5] gives a survey of these developments.…”

Non-genericity phenomena in ordered Fraïssé classes

“…In fact, as proved in [KPT05], if the group of automorphisms of a particular Fraïssé class K is extremely amenable, then there is a linear ordering on the Fraïssé limit of K that is preserved by all automorphisms. Moreover, the ordered limit is still Fraïssé , i.e., is a Fraïssé limit of a Fraïssé class.…”

“…This is partially due to the newly revealed connections between combinatorial properties of Fraïssé classes and algebraic, topological, and dynamical properties of the groups of automorphisms of their limits. One of the most astonishing links was established by Kechris, Pestov and Todorcevic in [KPT05] and displays a close relationship between Ramsey theory (a purely combinatorial area) and extreme amenability (a classical dynamical notion).…”

Automatic continuity of homomorphisms and fixed points on metric compacta

“…However, using the above results of Pestov and Glasner-Weiss and our automatic continuity result, we obtain the existence of such universal minimal metrisable flows for groups whose universal minimal flows are non-metrisable. Note that the universal minimal flows of the discrete groups considered in the corollary above are non-metrisable, as Kechris, Pestov and Todorcevic proved in [10] that the universal minimal flow of any locally compact, non-compact group is nonmetrisable.…”