Subgroups of the group of homeomorphisms of the Cantor space and a duality between a class of inverse monoids and a class of Hausdorff \'etale groupoids

TL;DR

This paper explores a duality between certain algebraic inverse monoids and topological groupoids, revealing how properties of one correspond to properties of the other, with applications to groups acting on the Cantor space.

Contribution

It establishes a non-commutative Stone duality linking Tarski inverse monoids with second countable Hausdorff étale groupoids, and characterizes their properties and associated groups.

Findings

Effective groupoids correspond to fundamental Tarski inverse monoids.

Minimal groupoids correspond to 0-simplifying Tarski inverse monoids.

Groups of units include Thompson groups and Krieger's ample groups.

Abstract

Under non-commutative Stone duality, there is a correspondence between second countable Hausdorff \'etale groupoids which have a Cantor space of identities and what we call Tarski inverse monoids: that is, countable Boolean inverse $\wedge$-monoids with semilattices of idempotents which are countable and atomless. Tarski inverse monoids are therefore the algebraic counterparts of the \'etale groupoids studied by Matui and provide a natural setting for many of his calculations. Under this duality, we prove that natural properties of the \'etale groupoid correspond to natural algebraic properties of the Tarski inverse monoid: effective groupoids correspond to fundamental Tarski inverse monoids and minimal groupoids correspond to $0$-simplifying Tarski inverse monoids. Particularly interesting are the principal groupoids which correspond to Tarski inverse monoids where every element is a finite join of infinitesimals and idempotents. Here an infinitesimal is simply a non-zero element with square zero. Such inverse monoids are natural infinite generalizations of finite symmetric inverse monoids. The groups of units of fundamental Tarski inverse monoids generalize the finite symmetric groups and include amongst their number the Thompson groups $G_{n,1}$ as well as the groups of units of what we term AF inverse monoids, Krieger's ample groups being examples. We characterize such groups as subgroups of particular kinds of the group of homeomorphisms of the Cantor space.