2022/10/24 by Julien Melleray, Melleray, Julien
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2210.13203
We investigate some properties of the clopen type semigroup of an action of a countable group on a compact, 0-dimensional, Hausdorff space X. We discuss some characterizations of dynamical comparison (most of which were already known in the metrizable case) in this setting; and prove that for a Cantor minimal action α of an amenable group the topological full group of α admits a dense, locally finite subgroup iff the corresponding clopen type semigroup is unperforated. We also discuss some properties of clopen type semigroups of the Stone-Čech compactifications and universal minimal flows of countable groups, and derive some consequences on generic properties in the space of minimal actions of a given countable group on the Cantor space.