Homology of gaussian groups

Dehornoy, Patrick; Lafont, Yves

Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 489-540.

Nous décrivons de nouvelles méthodes combinatoires fournissant des résolutions explicites du module trivial par des $\mathbb{Z}G$-modules libres lorsque $G$ est le groupe de fractions d’un monoïde possédant suffisamment de ppcm (“monoïde localement gaussien”), et donc, permettant de calculer l’homologie de $G$. Nos constructions s’appliquent en particulier à tous les groupes d’Artin–Tits de type de Coexeter fini. D’un point de vue technique, les démonstrations reposent sur les propriétés des ppcm dans un monoïde.

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 lest 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 Coexter type. Technically, the proofs rely on the properties of least common multiples in a monoid.

Classification : 20J06, 18G35, 20M50, 20F36
Keywords: free resolution, finite resolution, homology, contacting homotopy, braid groups, Artin groups
Mot clés : résolution libre, résolution finie, homologie, homotopie de contact, groupes de tresses, groupes d'Artin

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