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In this paper we consider those Fraïssé classes which admit companion classes in the sense of [KPT]. We find a necessary and sufficient condition for the automorphism group of the Fraïssé limit to be amenable and apply it to prove the non-amenability of the automorphism groups of the directed graph S(3) and the boron tree structure T. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $GL\left(V_{\infty }\right)$, where $V_{\infty }$ is the countably infinite-dimensional...

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 $S\left(G\right)$, the Samuel compactification, and $E\left(M\right(G\left)\right)$, 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...