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.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.05E15, 06A06, 20F36, 20F65, 51M20; 06A11Barycentric subdivision subdivides an n-cube into isometric metric simplices 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/ if and only if P has no short spindles.One way to view this conjecture is as an attempt to find something like the flag condition that tests whether a cube complex is CAT.0/. We highlight this perspective in Section 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/ if and only if P has no short spindles.Using Theorem A, we prove that the 5-string braid group, also known as the Artin group of type A 4 , is a CAT.0/ group. More precisely, we prove the following.Theorem B Let K be the Eilenberg-Mac Lane 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 A 4 or B 4 , the complex K is CAT.0/ and the group is a CAT.0/ group. When the group is of type D 4 , F 4 or H 4 , the complex K is not CAT.0/.

Dual euclidean Artin groups and the failure of the lattice property

“…When every pair of elements has a meet and a join, P is a lattice and when every subset has a meet and a join, it is a complete lattice. It turns out that a bounded graded poset P is a lattice iff P contains no bowties [7]. This makes it easy to show that certain subposets are also not lattices.…”

Non-crossing partitions

“…In fact, it generates the infinite cyclic center of B n . Brady-McCammond observed in [17] that this algebraic fact has a geometric counterpart:Ỹ n splits as a cartesian product of the real line R and another metric space. The R-factor insideỸ n points in the direction of the edges labelled by γ.…”

“…Brady-McCammond use a computer to analyse all loops in L shorter than 2π, and show that they are indeed shrinkable, thus establishing: Note that their proof is not computer assisted. The crucial improvement in the work of Haettel-Kielak-Schwer is to use the observation (present already in [17]), that the link L can be embedded into a spherical building, in the following way.…”

Factoring Euclidean isometries

“…To prove this, we begin by quoting a definition and a proposition from [BM10]. Proposition 9.3 (Lattice or bowtie).…”