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

Let $K$ be a 2–dimensional finite flag complex. We study the CAT(0) dimension of the ‘Bestvina–Brady group’, or ‘Artin kernel’, $Γ_{K}$ . We show that $Γ_{K}$ 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

Mathematical Subject Classification 2000

Primary: 20F67

Secondary: 57M20

References

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Publication

Received: 6 May 2002

Revised: 16 September 2002

Accepted: 12 October 2002

Published: 21 October 2002

Authors

John Crisp
Laboratoire de Topologie Université de Bourgogne UMR 5584 du CNRS BP 47 870 21078 Dijon France

Volume 2, issue 2 (2002)

John Crisp