[Paper Review] Sur les espaces test pour la moyennabilité

Yousef Al-Gadid, Brice Mbombo+1 more|arXiv (Cornell University)|Jun 25, 2010

Advanced Topology and Set Theory14 references3 citations

TL;DR

This paper establishes that the Hilbert cube and the Cantor space serve as universal test spaces for amenability, extreme amenability, and amenability at infinity in Polish and discrete countable groups. It proves that a Polish group is amenable if and only if every continuous action on the Hilbert cube admits an invariant probability measure, generalizing results by Bogatyi–Fedorchuk and Giordano–de la Harpe.

ABSTRACT

We observe that a Polish group $G$ is amenable if and only if every continuous action of $G$ on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the Cantor space can be used to detect amenability and extreme amenability of Polish non-archimedean groups as well as amenability at infinity of discrete countable groups. As corollary, the latter property can also be tested by actions on the Hilbert cube. These results generalize a criterion due to Giordano and de la Harpe.

Motivation & Objective

• To generalize the amenability criterion of Giordano and de la Harpe from discrete countable groups to all Polish groups.
• To show that the Hilbert cube is a universal test space for amenability of Polish groups, extending a result of Bogatyi and Fedorchuk.
• To establish that the Cantor space detects extreme amenability in nonarchimedean Polish groups.
• To prove that amenability at infinity of discrete countable groups can be tested via actions on the Cantor space or the Hilbert cube.
• To unify and extend existing criteria for amenability using the compactification of Samuel and dynamical systems on universal minimal spaces.

Proposed method

• Use the Samuel compactification $\mathcal{S}(G)$ of a Polish group $G$ as a universal dynamical model.
• Apply the Gelfand-Naimark duality to identify $C(\mathcal{S}(G))$ with the algebra of right uniformly continuous bounded functions on $G$.
• Construct invariant probability measures on the Hilbert cube $I^{\aleph_0}$ via approximation by finitely supported measures under continuous actions.
• Leverage the fact that $\mathcal{P}(D^{\aleph_0})$ is homeomorphic to $I^{\aleph_0}$ to transfer results between the Cantor space and the Hilbert cube.
• Use the notion of topological amenability via almost invariant nets of continuous maps $b^n: X \to \mathcal{P}(G)$ with $\|gb_x^n - b_{gx}^n\|_1 \to 0$.
• Apply the theorem of Keller to realize the space of probability measures on the Cantor space as homeomorphic to the Hilbert cube, enabling transfer of results.

Experimental results

Research questions

• RQ1 Can the Hilbert cube serve as a universal test space for amenability of all Polish groups, not just discrete countable ones?
• RQ2 Is the Cantor space a universal test space for extreme amenability in nonarchimedean Polish groups?
• RQ3 Can amenability at infinity of discrete countable groups be detected via actions on the Hilbert cube or the Cantor space?
• RQ4 Does the classical criterion of Giordano and de la Harpe extend beyond discrete countable groups to general Polish groups?
• RQ5 Is there a universal test space for extreme amenability in general Polish groups, given that the Hilbert cube fails?

Key findings

• A Polish group $G$ is amenable if and only if every continuous action of $G$ on the Hilbert cube $I^{\aleph_0}$ admits an invariant probability measure.
• The Cantor space $D^{\aleph_0}$ is a universal test space for extreme amenability of nonarchimedean Polish groups.
• A discrete countable group $G$ is amenable at infinity if and only if it admits a topologically amenable action on the Cantor space or on the Hilbert cube.
• The result of Giordano and de la Harpe on the Cantor space as a test for amenability of discrete countable groups extends to nonarchimedean Polish groups.
• The space of probability measures on the Cantor space, $\mathcal{P}(D^{\aleph_0})$, is homeomorphic to the Hilbert cube $I^{\aleph_0}$, enabling transfer of dynamical criteria.
• The Samuel compactification $\mathcal{S}(G)$ provides a universal framework for analyzing invariant measures and amenability in topological groups.