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.Comment: 29 pages ; v2 : cross-references updated ; v3 : typos corrected;
final version due to appear in Journal of Combinatorial Algebr
