Topological full groups of etale groupoids

TL;DR

This survey reviews recent advances in the study of topological full groups of etale groupoids, focusing on their algebraic, geometric, and analytic properties derived from dynamical systems on the Cantor set.

Contribution

It consolidates recent results on properties like simplicity, amenability, and cohomological aspects of topological full groups associated with etale groupoids.

Findings

Simplicity of commutator subgroups established

Conditions for amenability and the Haagerup property analyzed

Connections between homology groups and K-theory explored

Abstract

This is a survey of the recent development of the study of topological full groups of etale groupoids on the Cantor set. Etale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations. Minimal Z-actions, minimal Z^N-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 etale groupoids, groupoid C*-algebras and their K-groups are also investigated.