Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products

Abstract

We compute:

• the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), of Bestvina–Brady groups of type FP, and of graph products of groups,
• the $L^2$-Betti numbers of Bestvina–Brady groups of type FP over $\mathbb{Q}$, and of graph products of groups,
• the weighted $L^2$-Betti numbers of graph products of Coxeter groups.

In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.