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

TL;DR

This paper computes cohomology, L^2-Betti numbers, and weighted L^2-Betti numbers for Artin groups, Bestvina-Brady groups, and graph products, providing new insights into their algebraic and topological properties.

Contribution

It offers explicit cohomology and Betti number calculations for these complex group classes, extending previous results and including weighted invariants for Coxeter group graph products.

Findings

Cohomology with group ring coefficients computed for Artin, Bestvina-Brady, and graph product groups.

L^2-Betti numbers determined for Bestvina-Brady and graph product groups.

Weighted L^2-Betti numbers calculated for graph products of Coxeter groups.

Abstract

We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti numbers of Bestvina-Brady groups 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.(However, for graph products of Coxeter groups this proviso is unnecessary.)