The polycyclic inverse monoids and the Thompson groups revisited

TL;DR

This paper explores the connections between polycyclic inverse monoids, Cuntz inverse monoids, and Thompson groups, revealing new structural insights and their relationships with Cuntz $C^*$-algebras and Cantor algebras.

Contribution

It establishes that the Thompson group $G_{n,1}$ is the group of units of the Cuntz inverse monoid and relates it to the automorphisms of an $n$-ary Cantor algebra, providing new structural understanding.

Findings

Thompson group $G_{n,1}$ is the group of units of the Cuntz inverse monoid.

The Cuntz inverse monoid is the tight completion of the polycyclic inverse monoid.

The associated groupoid corresponds to the Cuntz $C^*$-algebra.

Abstract

We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid $C_{n}$ called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid $P_{n}$. The \'etale topological groupoid associated with $C_{n}$ under non-commutative Stone duality is the usual groupoid associated with the corresponding Cuntz $C^{\ast}$-algebra. We then show that the group $G_{n,1}$ is also the group of automorphisms of a specific $n$-ary Cantor algebra: this $n$-ary Cantor algebra is constructed first as the monoid of total maps of a restriction semigroup \`a la Statman and then in terms of labelled trees \`a la Higman.