Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem

Abstract

For G a closed subgroup of S ∞ , we provide a precise combinatorial characterization of when the universal minimal flow M (G) is metrizable. In particular, each such instance fits into the framework of metrizable flows developed in [KPT] and [NVT]; as a consequence, each G with metrizable universal minimal flow has the generic point property, i.e. every minimal G-flow has a point whose orbit is comeager. This solves the Generic Point Problem raised in [AKL] for closed subgroups of S ∞ . space of structures of the form K, < , where < is a linear ordering of N. We endow X K LO with the topology whose basic open neighborhoods are of the form N (L) = { K, < ∈ X K LO : <| [k] = L}, where L is some linear ordering of [k] = {1, 2, ..., k}. With this topology, X K LO is compact and metrizable. We let G act on X K LO via K, < · g = K, < g where (m < g n) iff (g(m) < g(n)). This turns X K LO into a G-flow. When K is a Fraïssé structure, < 0 ∈ X K LO with K,

Automorphism groups and Ramsey properties of sparse graphs

“…In fact, the converse is also true: if is metrisable, then there is a comeagre ‐orbit on and the stabiliser of a point in this orbit is extremely amenable and co‐precompact in . This is proved by Zucker in and, independently, by Ben Yaacov, Melleray and Tsankov in . (The latter builds on work in and proves the result for arbitrary Polish groups .…”

“…Remarks (1)By the above result and [, Theorem 1.2], we know that some structure in has infinite Ramsey degree (as defined in Section 2.2). In fact, the argument in the proof of part (1) above shows that, in the notation used there, has infinite Ramsey degree in .…”

“…Using , we then have the following corollary, answering Question 1.5 in : Corollary There is a closed, oligomorphic permutation group on a countable set whose universal minimal flow is not metrisable.…”

“…The following (from [, Theorem 4]; see also [, Proposition 5.5]) gives a criterion for minimality of the ‐flow . It relates to the notion of Expansion Property defined as follows: Definition Let be a reasonable class of expansions of the amalgamation class .…”

“…We then have the following version of [, Theorem 10.8; , Theorem 5]. See also [, Theorem 5.7]. Theorem Let be first‐order languages with relational.…”