Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs

TL;DR

This paper characterizes when automorphism groups of countable homogeneous directed graphs are amenable and uniquely ergodic, using topological dynamics and extending existing techniques to new classes of structures.

Contribution

It provides a complete classification of amenability and unique ergodicity for automorphism groups of these graphs, including new methods for non-amenable cases.

Findings

Identifies which automorphism groups are amenable.

Determines unique ergodicity for most cases.

Leaves open the semi-generic complete multipartite case.

Abstract

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the "semi-generic" complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of [3], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in [26]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context of countable homogeneous directed graphs is given in [10] and the papers cited therein.