On the CAT(0) dimension of 2–dimensional Bestvina–Brady groups

Abstract

Let $K$ be a 2–dimensional finite flag complex. We study the CAT(0) dimension of the ‘Bestvina–Brady group’, or ‘Artin kernel’, ${\Gamma }_K$. We show that ${\Gamma }_K$ has CAT(0) dimension 3 unless $K$ admits a piecewise Euclidean metric of non-positive curvature. We give an example to show that this implication cannot be reversed. Different choices of $K$ 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.