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

### Patrick Dehornoy

Université de Caen, France

## 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.

## Cite this article

Patrick Dehornoy, Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC. J. Comb. Algebra 1 (2017), no. 2, pp. 185–228

DOI 10.4171/JCA/1-2-3