Sala: 243-A
Palestrante: Brice Rodrigue Mbombo, IME-USP
Título: Extreme amenability of the isometry group of the Urysohn space
Resumo: The definition of an extremely amenable group is obtained from the classical definition of an amenable group by removing the two underlined words of the definition: A topological group $G$ is amenable if every continuous $\underline{\text{affine}}$ action of $G$ on compact $\underline{\text{convex}}$ set $X$ admits a fixed point: for some $\xi\in X$ and all $g\in G$, one has $g\xi=\xi$.
The existence of extremely amenable semigroups was proved by Granirer in $1967$. But it was at first unclear if extremely amenable topological groups existed at all. The first example of this kind was done by Herer and Christensen in $1975$. Some further examples known to date include:
- $\mathcal{U}(\ell^{2})$, equipped with strong operator topology (Gromov-Milman, $1984$);
- $Aut\,(\mathbb{Q},\leq)$ with the topology of simple convergence (Pestov, $1998$);
- $Iso(\mathbb{U})$ where $\mathbb{U}$ is the universal Urysohn space (Pestov, $2002$).
In this talk, we will provide a short history(motivations) of the subject, recall the Kat\v etov construction of the Urysohn space $\mathbb{U}$ and give all details of Pestov Proof of extreme amenability of the group $Iso(\mathbb{U})$.