Stabilizers and maximal subgroups of topological full groups

Abstract

We study closed-set stabilizers in topological full groups of minimal effective ample groupoids with Cantor unit space. For almost finite or purely infinite groupoids, we extend the maximal stabilizer theorem of Grigorchuk and Vorobets to every subgroup containing the commutator subgroup. For a nonempty closed nowhere dense set, maximality of its stabilizer is equivalent to minimality of its stabilizer action, minimality of the regular reduction, and primeness of a canonical wide open subgroupoid. Without assuming almost finiteness or pure infiniteness, we prove spatial rigidity for stabilizers acting minimally on infinite closed nowhere dense sets. Their isomorphism types determine the ambient Kakutani equivalence class and, when Hausdorff, the effective reduction. We also realize every AF groupoid with compact unit space as the full restriction, with its relative topology, of any minimal effective ample groupoid with Cantor unit space to a closed nowhere dense subset. Combining rigidity and realization yields continuum many pairwise nonisomorphic stabilizers in every subgroup containing the commutator subgroup; these stabilizers are maximal when the ambient groupoid is almost finite or purely infinite. The realization also gives explicit AF subquotients of the ambient reduced groupoid $C^*$-algebra, including the CAR algebra, with compatibility on the canonical diagonal subalgebras.