Continuity of the stabilizer map and irreducible extensions

2023/02/06 by Adrien Le Boudec, Boudec, Adrien Le, Todor Tsankov +1
Mathematics · #06E15 #54H15 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Primary: 37B05. Secondary: 22D12

paper · pdf · doi:10.48550/arxiv.2302.03083

openalex publication_date 2023/02/06 · openalex created_date 2023/02/10 · openalex updated_date 2026/07/28

Abstract

Let G be a locally compact group. For every G-flow X, one can consider the stabilizer map x ↦ G_x, from X to the space 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 S_G(X) in the space Sub(G), which generalizes the notion of stabilizer uniformly recurrent subgroup associated to a minimal G-flow introduced by Glasner and Weiss.

Citations

• Boundaries of reduced C*-algebras of discrete groups

Related

• Realizing uniformly recurrent subgroups
• Universal minimal flows of extensions of and by compact groups
• Absorptive Continuous R-group Actions on Locally Compact Spaces
• Veech's Theorem of G acting freely on G^\textrmLUC and Structure Theorem of a.a. flows
• An extension criterion for lattice actions on the circle