Topological full groups of Étale groupoids

Abstract

This is a survey of the recent development of the study of topological full groups of étale groupoids on the Cantor set. Étale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations.Minimal ℤ-actions, minimal ℤN-actions and one-sided shifts of finite type are basic examples. We are interested in algebraic, geometric and analytic properties of topological full groups. More concretely, we discuss simplicity of commutator subgroups, abelianization, finite generation, cohomological finiteness properties, amenability, the Haagerup property, and so on. Homology groups of étale groupoids, groupoid C∗-algebras and their K-groups are also investigated.