## Algebraic & Geometric Topology

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

John Crisp

#### Abstract

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

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

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

https://projecteuclid.org/euclid.agt/1513882746

Digital Object Identifier
doi:10.2140/agt.2002.2.921

Mathematical Reviews number (MathSciNet)
MR1936975

Zentralblatt MATH identifier
1055.20036

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

#### Citation

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. https://projecteuclid.org/euclid.agt/1513882746

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