# Journal of Combinatorial Algebra

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**Volume 1, Issue 2, 2017, pp. 185–228**

**DOI: 10.4171/JCA/1-2-3**

Published online: 2017-04-06

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

Patrick Dehornoy^{[1]}(1) Université de Caen, France

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.

*Keywords: *Artin–Titsmonoid,Artin–Tits group, gcd-monoid, enveloping group, word problem, multifraction, reduction, 3-Ore condition, type FC, embeddability, normal form, van Kampen diagram.

Dehornoy Patrick: Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC. *J. Comb. Algebra* 1 (2017), 185-228. doi: 10.4171/JCA/1-2-3