Braids, posets and orthoschemes

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.