Wednesday, November 14, 2018, 11:00

Prague – IM AS CR, Zitna 25, seminar room, front building, third floor

Speaker: Stefan Geschke

Title: There are no universal minimal metric flows for countable discrete groups

Abstract:

Let $G$ be a topological group. A $G$-flow is a compact space $X$ together with a continuous action of $G$. The morphisms between $G$-flows are continuous maps that respect the group action. A flow is minimal if it has no proper (nonempty) subflows. A flow $X$ is universal in a class $\mathcal C$ if it is in $\mathcal C$ and for every flow $Y \in \mathcal C$ there is an epimorphism from $X$ onto $Y$. Using Fürstenberg’s structure theorem for distal flows, Foreman and Beleznay showed that there are no universal minimal metric $\mathbb Z$-flows.

Every group $G$ acts in a natural way on the space $2^G$. Gao and Jackson showed that for every countable discrete group $G$, $2^G$ has a perfect set of minimal subflows. We show that this implies that there are no universal minimal metric flows for any countable discrete group $G$.