Work

Continuity of the stabilizer map and irreducible extensions

Year: 2024

Type: article

Abstract: Let G be a locally compact group. For every G -flow X , one can consider the stabilizer map x \mapsto G_{x} , from X to the space \mathrm{Sub}(G) of closed subgroups of G . This map is not continuous in general. We prove that if one passes from X to the universal irreducible extension of X , the stabilizer map becomes continuous. This result provides, in particular, a common generalization of a theorem of Frolík (that the set of fixed points of a homeomorphism of an extremally disconnected compact space is open) and a theorem of Veech (that the action of a locally compact group on its greatest ambit is free). It also allows to naturally associate to every G -flow X a stabilizer G -flow \mathrm{S}_{G}(X) in the space \mathrm{Sub}(G) , which generalizes the notion of stabilizer uniformly recurrent subgroup associated to a minimal G -flow introduced by Glasner and Weiss.

Source: Commentarii Mathematici Helvetici

DOI: 10.4171/cmh/583

Authors: Adrien Le Boudec, Todor Tsankov

Institutions: École Normale Supérieure de Lyon, Centre National de la Recherche Scientifique, Unité de Mathématiques Pures et Appliquées, Lyon 1 Université, Institut Universitaire de France

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Agence Nationale de la Recherche

Award: Group Actions and Model Theory