Algebraic & Geometric Topology

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

John Crisp

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

Article information

Algebr. Geom. Topol., Volume 2, Number 2 (2002), 921-936.

Received: 6 May 2002
Revised: 16 September 2002
Accepted: 12 October 2002
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 57M20: Two-dimensional complexes

nonpositive curvature dimension flag complex Artin group


Crisp, John. On the CAT(0) dimension of 2–dimensional Bestvina–Brady groups. Algebr. Geom. Topol. 2 (2002), no. 2, 921--936. doi:10.2140/agt.2002.2.921.

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