Generic dense free subgroups of the isometry group of the Urysohn space are NSS

The isometry group of the bounded Urysohn space, $G=\mathrm{I}\mathrm{s}\mathrm{o}(\text{U}1)$ is a central object in the study of Polish groups and topological dynamics. It is known that generic sequences in $G$ generate algebraically free dense subgroups. In this paper, we show that such generic free subgroups exhibit strong geometric rigidity. Specifically, we prove that for a comeager set of sequences generating dense free subgroups $F\le G$, every non-trivial element $h\in F$ acts with maximal metric displacement, satisfying ${\sup }_{n\in \mathbb{N}}d(h^n(x),x)=1$ for every $x\in \text{U}1$. As a consequence, these generic subgroups satisfy the \emph{no small subgroup} ($\text{nss}$) property. We note that the method naturally extends to the full isometry group $\mathrm{I}\mathrm{s}\mathrm{o}(\mathbb{U})$ of the classical Urysohn space.