Higher dimensional generalizations of the Thompson groups

Abstract

Motivated by the goal of generalizing the Cuntz-Krieger algebras, and making heavy use of the pioneering work of Robertson and Steger, Kumjian and Pask generalized the notion of a directed graph to what they termed a ‘higher rank graph’. The C⁎-algebras constructed from such higher rank graphs have proved to be highly interesting. Higher rank graphs are, in fact, a class of cancellative categories and so the one-vertex higher rank graphs are therefore a class of cancellative monoids — in this paper we call them k-monoids. Finite direct products of free monoids belong to this family but there are many examples not of this form. It is well-known that the classical Thompson groups arise from the free monoids, and it is easy to show that Brin's higher dimensional Thompson groups arise from finite direct products of free monoids. In this paper, we generalize these constructions and show how to construct a family of new groups with simple commutator subgroups from arbitrary aperiodic k-monoids thereby subsuming all the above examples. Although we give simple direct constructions of our groups, a key feature of our paper is that we also show that they arise as topological full groups of the usual étale groupoids associated with k-monoids. In this way, our groups are naturally associated with C⁎-algebras. The question of the mutual isomorphisms amongst this family of groups appears to be a delicate one and is not handled here.