Amenability and Unique Ergodicity of Automorphism Groups of Fra\"iss\'e Structures

TL;DR

This paper establishes a criterion for the amenability of automorphism groups of Fra"iss"e structures, demonstrates non-amenability for specific structures, and addresses the Unique Ergodicity-Generic Point problem with a counterexample involving infinite-dimensional vector spaces.

Contribution

It provides a necessary and sufficient condition for amenability, applies it to specific structures to show non-amenability, and offers a negative answer to a longstanding ergodicity problem.

Findings

Automorphism groups of certain structures are non-amenable.

The unique invariant measure on the universal minimal flow may not be supported on the generic orbit.

A counterexample to the Unique Ergodicity-Generic Point problem is constructed using infinite-dimensional vector spaces.

Abstract

In this paper we provide a necessary and sufficient condition for the amenability of the automorphism group of Fra\"iss\'e structures 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.