Coherent presentations of structure monoids and the Higman-Thompson groups

TL;DR

This paper develops a method to construct presentations of structure monoids and groups from coherent categorifications of algebraic varieties, leading to new realizations of higher Thompson and Higman-Thompson groups.

Contribution

It introduces a general framework linking coherent categorifications to presentations of structure groups, extending previous results and providing new group realizations.

Findings

Realized higher Thompson groups as structure groups.

Derived presentations of Higman-Thompson groups via categorification.

Generalized Mac Lane's coherence conditions for higher-order structures.

Abstract

Structure monoids and groups are algebraic invariants of equational varieties. We show how to construct presentations of these objects from coherent categorifications of equational varieties, generalising several results of Dehornoy. We subsequently realise the higher Thompson groups $F_{n,1}$ and the Higman-Thompson groups $G_{n,1}$ as structure groups. We go on to obtain presentations of these groups via coherent categorifications of the varieties of higher-order associativity and of higher-order associativity and commutativity, respectively. These categorifications generalise Mac Lane's pentagon and hexagon conditions for coherently associative and commutative bifunctors.