Higher dimensional generalizations of the Thompson groups

TL;DR

This paper constructs a new family of higher-dimensional groups extending Thompson groups, using higher rank graphs and inverse semigroups, revealing their algebraic structure and connections to monoids.

Contribution

It introduces a novel method to build higher-dimensional Thompson-like groups from higher rank graphs and inverse semigroups, expanding the understanding of their algebraic properties.

Findings

Constructed groups have simple commutator subgroups.

Established a link between higher rank graphs and generalized Thompson groups.

Demonstrated these groups as higher-dimensional analogs of classical Thompson groups.

Abstract

We show how to construct a family of groups with simple commutator subgroups from aperiodic 1-vertex, finitely aligned higher rank graphs (which are, in fact, a class of cancellative monoids). Inverse semigroups form the intermediary between these cancellative monoids and the family of groups we are interested in. These groups can naturally be viewed as higher-dimensional generalizations of the classical Thompson groups since the finite direct products of free monoids are examples of the appropriate 1-vertex higher rank graphs.