Amenability and unique ergodicity of automorphism groups of Fraïssé structures
Volume 226 / 2014
Abstract
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 $\mathbf {S}(3)$ and the boron tree structure $\mathbf {T}$. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel–Kechris–Lyons [AKL]. By considering $\mathrm {GL}(\mathbf {V}_\infty )$, where $\mathbf {V}_\infty $ 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 $\mathrm {GL}(\mathbf {V}_\infty )$ is not supported on the generic orbit.
