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Fraïssé structures and a conjecture of Furstenberg. (English) Zbl 07088822
Summary: We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between \(S(G)\), the Samuel compactification, and \(E(M(G))\), the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of \(G= S_\infty\), leading us to define and investigate several new types of ultrafilters on a countable set.

MSC:
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
05C63 Infinite graphs
03E05 Other combinatorial set theory
22F50 Groups as automorphisms of other structures
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