Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC

TL;DR

This paper introduces a new convergent rewrite system for Artin-Tits groups satisfying the 3-Ore condition, providing a solution to the Word Problem and unique element representations.

Contribution

It extends Ore's theorem to a broader class of groups using a novel rewrite system under the 3-Ore condition.

Findings

The rewrite system R(M) is convergent under the 3-Ore condition.

Unique representations of elements in U(M) are established.

Universal shapes for van Kampen diagrams are identified.

Abstract

We describe a new approach to the Word Problem for Artin-Tits groups and, more generally, for the enveloping group U(M) of a monoid M in which any two elements admit a greatest common divisor. The method relies on a rewrite system R(M) that extends free reduction for free groups. Here we show that, if M satisfies what we call the 3-Ore condition about common multiples, what corresponds to type FC in the case of Artin-Tits monoids, then the system R(M) is convergent. Under this assumption, we obtain a unique representation result for the elements of U(M), extending Ore's theorem for groups of fractions and leading to a solution of the Word Problem of a new type. We also show that there exist universal shapes for the van Kampen diagrams of the words representing 1.