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 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.