Let be a 2–dimensional finite flag complex. We study the CAT(0) dimension of the ‘Bestvina–Brady group’, or ‘Artin kernel’, . We show that has CAT(0) dimension 3 unless admits a piecewise Euclidean metric of non-positive curvature. We give an example to show that this implication cannot be reversed. Different choices of lead to examples where the CAT(0) dimension is 3, and either (i) the geometric dimension is 2, or (ii) the cohomological dimension is 2 and the geometric dimension is not known.
"On the CAT(0) dimension of 2–dimensional Bestvina–Brady groups." Algebr. Geom. Topol. 2 (2) 921 - 936, 2002. https://doi.org/10.2140/agt.2002.2.921