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

Abstract

Under non-commutative Stone duality, there is a correspondence between second countable Hausdorff étale 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 étale groupoids studied by Matui and provide a natural setting for many of his calculations. Under this duality, we prove that natural properties of the étale 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.