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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 uniformly recurrent subgroups (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. [ABSTRACT FROM AUTHOR]

We show that every bounded automaton group can be embedded in a finitely generated, simple amenable group. The proof is based on the study of the topological full groups associated to the Schreier dynamical system of the mother groups. We also show that if 풢 is a minimal étale groupoid with unit space the Cantor set, the group [[g]]t generated by all torsion elements in the topological full group has simple commutator subgroup. [ABSTRACT FROM AUTHOR]

We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure. [ABSTRACT FROM AUTHOR]