Generic representations of abelian groups and extreme amenability

If $G$ is a Polish group and $\Gamma$ is a countable group, denote by \Hom(\Gamma, G) the space of all homomorphisms $\Gamma \to G$. We study properties of the group \cl{\pi(\Gamma)} for the generic \pi \in \Hom(\Gamma, G), when $\Gamma$ is abelian and $G$ is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on $\Gamma$, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic \cl{\pi(\Gamma)}; in the other two, we show that the generic \cl{\pi(\Gamma)} is extremely amenable. We also show that if $\Gamma$ is torsion-free, the centralizer of the generic $\pi$ is as small as possible, extending a result of King from ergodic theory.Comment: Version

Cybercities: Mediated Public Open Spaces - A Matter of Interaction and Interfaces.

In the near past, sources of information about public open spaces were: people, the place itself and historical archives. Accordingly, the information could be obtained by interviewing the visitors, by reading some poorly equipped signs on monuments or by research in libraries. Today, a new source appeared: The place itself covers its own information by the mean of the growing of the ICT (Information Communication Technologies). In addition, the information can be personalised in a way each people can access it individually. Ten years ago, a left-over newspaper on a park bench was a compact piece of information. Today, the newspaper resides on a smartphone in our pockets. In the future, the park bench will still be there, but dramatically changed to an IoT (Internet of things) object, bringing information to the people. Therefore, there is the need to re-think the park bench as an interface. A simple, fundamental point is: the quality of the interface rules the quality of the information. With a special focus on the latter, this chapter discusses how the classical model of the city is enhanced with the senseable city concept and how digital information influences, adopts, transforms and re-configures different objects in urban areas