A commutator lemma for confined subgroups and applications to groups acting on rooted trees

A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $\operatorname{Sub}(G)$ of subgroups of $G$. We prove a commutator lemma for confined subgroups. For groups of homeomorphisms, this provides the exact analogue for confined subgroups (hence in particular for URSs) of the classical commutator lemma for normal subgroups: if $G$ is a group of homeomorphisms of a Hausdorff space $X$ and $H$ is a confined subgroup of $G$, then $H$ contains the derived subgroup of the rigid stabilizer of some open subset of $X$. We apply this commutator lemma in the setting of groups acting on rooted trees. We prove a theorem describing the structure of URSs of weakly branch groups and of their non-topologically free minimal actions. Among the applications of these results, we show: 1) if $G$ is a finitely generated branch group, the $G$-action on $\partial T$ has the smallest possible growth among all faithful $G$-actions; 2) if $G$ is a finitely generated branch group, then every embedding from $G$ into a group of homeomorphisms of strongly bounded type (e.g. a bounded automaton group) must be spatially realized; 3) if $G$ is a finitely generated weakly branch group, then $G$ does not embed into the group IET of interval exchange transformations.