Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Andy Zucker¹

¹ Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213, U.S.A.

Fundamenta Mathematicae, Tome 226 (2014) no. 1, pp. 41-61

Résumé

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.

Détail

DOI : 10.4064/fm226-1-3

Mots-clés : paper consider those fra classes which admit companion classes sense kpt necessary sufficient condition automorphism group fra limit amenable apply prove non amenability automorphism groups directed graph mathbf boron tree structure nbsp mathbf provide negative answer unique ergodicity generic point problem angel kechris lyons akl considering mathrm mathbf infty where mathbf infty countably infinite dimensional vector space finite field unique invariant measure universal minimal flow mathrm mathbf infty supported generic orbit