Abstract: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.
Keywords: Fra\"\i ss\'e structures; enveloping semigroups; universal minimal flow
DOI: DOI 10.14712/1213-7243.2015.276
AMS Subject Classification: 37B05 05C63 03E05 22F50