2022/05/12 by Steven Hurder, Hurder, Steven, Olga Lukina +1
Mathematics · #22F10 #37A15 #37B05 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Primary: 20E18 #Secondary: 57S10
paper · pdf · doi:10.48550/arxiv.2205.06285
openalex publication_date 2022/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, which means that the set of points with non-trivial stabilizer has positive measure. In this paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. In particular, this gives a new proof that a minimal equicontinuous Cantor action by a nilpotent group has no essential holonomy. We also show that the property of having essential holonomy is preserved under return equivalence and continuous orbit equivalence of minimal equicontinuous Cantor actions. Finally, we give examples to show that the assumption on the action that it is locally quasi-analytic is necessary.