2020/06/15 by Adrien Le Boudec, Nicolás Matte Bon, Boudec, Adrien Le +1 · 1 citation
Chemistry · Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Coordination Chemistry and Organometallics #FOS: Mathematics #Fluorine in Organic Chemistry #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2006.08677
openalex publication_date 2020/06/15 · openalex created_date 2023/07/05 · openalex updated_date 2026/07/28
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 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 ∂ 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.