3 problems

Let $\mathbb U$ be Urysohn's universal metric space and let $Iso(\mathbb U)$ denote its isometry group. For $\varphi\in Iso(\mathbb U)$, write $Fix(\varphi)$ for the set of fixed p…

Let $\mathbb{U}$ be the Urysohn space, and let $\mathbb{H}$ be Hall's universal countable locally finite group. The group $\operatorname{Iso}(\mathbb{U})$ is the isometry group of…

Let $\bU_1$ be the bounded Urysohn space, and let $\Iso(\bU_1)$ denote its isometry group. Simplicity conjecture. The group $\Iso(\bU_1)$ is simple. If true, this would imply that…