#### Volume 10, issue 4 (2010)

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Braids, posets and orthoschemes

### Tom Brady and Jon McCammond

Algebraic & Geometric Topology 10 (2010) 2277–2314
##### Abstract

In this article we study the curvature properties of the order complex of a bounded graded poset under a metric that we call the “orthoscheme metric”. In addition to other results, we characterize which rank $4$ posets have $CAT\left(0\right)$ orthoscheme complexes and by applying this theorem to standard posets and complexes associated with four-generator Artin groups, we are able to show that the $5$–string braid group is the fundamental group of a compact nonpositively curved space.

##### Keywords
braid group, Artin group, poset, orthoscheme, nonpositive curvature
##### Mathematical Subject Classification 2000
Primary: 05E15, 06A06, 20F36, 20F65, 51M20
Secondary: 06A11