Braids, posets and orthoschemes

TL;DR

This paper investigates the curvature properties of order complexes of graded posets using the orthoscheme metric, characterizing certain posets with nonpositive curvature and linking braid groups to nonpositively curved spaces.

Contribution

It introduces the orthoscheme metric for order complexes and characterizes rank 4 posets with CAT(0) properties, connecting braid groups to nonpositively curved spaces.

Findings

Characterization of rank 4 posets with CAT(0) orthoscheme complexes

The 5-string braid group is the fundamental group of a compact nonpositively curved space

Application of the theory to complexes associated with four-generator Artin groups

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 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.