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Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Volume 226 / 2014

Andy Zucker Fundamenta Mathematicae 226 (2014), 41-61 MSC: Primary 37B05; Secondary 03C15, 03E02, 03E15, 05D10, 22F10, 22F50, 43A07, 54H20. DOI: 10.4064/fm226-1-3

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.

Authors

  • Andy ZuckerDepartment of Mathematical Sciences
    Carnegie Mellon University
    Pittsburgh, PA 15213, U.S.A.
    e-mail

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