### Brice Rodrigue Mbombo: Amenability test spaces for Polish groups

Data: segunda-feira, 23 de setembro de 2013, às 14h

Sala: 267-A

Palestrante: Brice Rodrigue Mbombo, IME-USP

Título: Amenability test spaces for Polish groups

Resumo: A topological group $G$ is amenable if every continuous action of  $G$ on a compact space $X$ admits an invariant Borel probability measure.

A compact space $K$ is an amenability test space for a class $\mathcal{C}$ of topological groups if a group $G \in \mathcal{C}$ is amenable if and only if every continuous action of $G$ on $K$ admits an invariant Borel probability measure. De la Harpe and Giordano (C. R. Acad. Sci. Paris 324(1997), 1255–1258), answering a question of Grigorchuk, had proved that the Cantor space $2^{\aleph_{0}}$ is an amenability test space for discrete countable groups. Bogatyi and Fedorchuk (Topol. Methods Nonlinear Anal. 29(2007), 383–401) had obtained the same conclusion for the Hilbert cube $[0,1]^{\aleph_{0}}$. We observe that the result of Bogatyi and Fedorchuk remain true for the class of Polish groups. 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 generalise a criterion due to Giordano and de la Harpe.

At the same time, we do not know whether there exists a compact metrizable test space $K$ for detecting extreme amenability of Polish groups, in other words, having the property that a Polish group $G$ has a fixed point in every compact space upon which it acts continuously whenever it has fixed point whenever it acts continuously on $K$. A separable compact $K$ with this property exists for cardinality considerations.

In this couple of talks, we will provide an history introduction to the notion of amenability for groups before speaking  about amenability test spaces for Polish groups.