Groups, Geometry, and Dynamics


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Volume 6, Issue 3, 2012, pp. 485–531
DOI: 10.4171/GGD/164

Published online: 2012-08-16

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

Michael W. Davis[1] and Boris Okun[2]

(1) Ohio State University, Columbus, USA
(2) University of Wisconsin at Milwaukee, USA

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.

Keywords: Artin group, Bestvina–Brady group, building, Coxeter group, graph product, right-angled Artin group, $L^2$-Betti number, weighted $L^2$-cohomology

Davis Michael, Okun Boris: Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products. Groups Geom. Dyn. 6 (2012), 485-531. doi: 10.4171/GGD/164