#### Volume 2, issue 2 (2002)

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

### John Crisp

Algebraic & Geometric Topology 2 (2002) 921–936
 arXiv: math.GR/0211130
##### 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.

##### Keywords
nonpositive curvature, dimension, flag complex, Artin group
Primary: 20F67
Secondary: 57M20