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

Abstract

We describe a new approach to the word problem for Artin–Tits groups and, more generally, for the enveloping group $\mathcal{U}(M)$ of a monoid $M$ in which any two elements admit a greatest common divisor. The method relies on a rewrite system ${\mathcal{R}}_M$ 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 ${\mathcal{R}}_M$ is convergent. Under this assumption, we obtain a unique representation result for the elements of $\mathcal{U}(M)$, 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.