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 posets have 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 –string braid group is the fundamental group of a compact nonpositively curved space.
"Braids, posets and orthoschemes." Algebr. Geom. Topol. 10 (4) 2277 - 2314, 2010. https://doi.org/10.2140/agt.2010.10.2277