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Homology of Gaussian groups. (English) Zbl 1100.20036
Summary: We describe new combinatorial methods for constructing explicit free resolutions of \(\mathbb{Z}\) by \(\mathbb{Z} G\)-modules when \(G\) is a group of fractions of a monoid where enough least common multiples exist (“locally Gaussian monoid”), and therefore, for computing the homology of \(G\). Our constructions apply in particular to all Artin-Tits groups of finite Coxeter type. Technically, the proofs rely on the properties of least common multiples in a monoid.

MSC:
20J05 Homological methods in group theory
20M05 Free semigroups, generators and relations, word problems
20F36 Braid groups; Artin groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20M50 Connections of semigroups with homological algebra and category theory
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