Braids, posets and orthoschemes

Abstract

Abstract. In this article we study the curvature properties of the order complex of a graded poset under a metric that we call the “orthoscheme metric”. In addition to other results, we characterize which rank 4 posets have CAT(0) orthoscheme complexes and by applying this the-orem 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. Barycentric subdivision subdivides an n-cube into isometric metric sim-plices called orthoschemes. We use orthoschemes to turn the order complex of a graded poset P into a piecewise Euclidean complex K that we call its orthoscheme complex. Our goal is to investigate the way that combinatorial properties of P interact with curvature properties of K. More specifically, we focus on combinatorial configurations in P that we call spindles and conjecture that they are the only obstructions to K being CAT(0). Poset Curvature Conjecture. The orthoscheme complex of a bounded graded poset P is CAT(0) iff P has no short spindles. One way to view this conjecture is as an attempt to extend to a broader context the flag condition that tests whether a cube complex is CAT(0). We highlight this perspective in §7. Our main theorem establishes the conjecture for posets of low rank. Theorem A. The orthoscheme complex of a bounded graded poset P of rank at most 4 is CAT(0) iff P has no short spindles. Using Theorem A, we prove that the 5-string braid group, also known as the Artin group of type A4, is a CAT(0) group. More precisely, we prove the following. Theorem B. Let K be the Eilenberg-MacLane space for a four-generator Artin group of finite type built from the corresponding poset of W-noncrossing partitions and endowed with the orthoscheme metric. When the group is of type A4 or B4, the complex K is CAT(0) and the group is a CAT(0) group. When the group is of type D4, F4 or H4, the complex K is not CAT(0). The article is structured as follows. The initial sections recall basic results about posets, complexes and curvature, followed by sections establishing the key properties of orthoschemes, orthoscheme complexes and spindles. The final sections prove our main results and contain some concluding remarks.