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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 twoelements admit a greatest common divisor. The method relies on a rewrite systemR(M) that extends free reduction for free groups. Here we show that, if Msatisfies what we call the 3-Ore condition about common multiples, whatcorresponds 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 resultfor the elements of U(M), extending Ore's theorem for groups of fractions andleading to a solution of the Word Problem of a new type. We also show thatthere exist universal shapes for the van Kampen diagrams of the wordsrepresenting 1.

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