Amenability and unique ergodicity of automorphism groups of Fraisse structures

In this paper we consider those Fraisse classes which admit companion classes in the sense of [KPT]. We find a necessary and sufficient condition for the automorphism group of the Fraisse 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(V-infinity), where V-infinity is the countably infinite-dimensional vector space over a finite field F-q, we show that the unique invariant measure on the universal minimal flow of GL(V-infinity) is not supported on the generic orbit.