The planar pure braid group is a diagram group

Abstract

A planar pure braid consists of $n$ descending smooth arcs, each connecting a point on one horizontal line $\ell_{1}$ to a point on a horizontal line $\ell_{2}$, which is required to be directly below the first point. Two arcs are allowed to cross, but no threefold intersections are allowed. The set $\Gamma_{n}$ of all planar pure braids on $n$ strands is a group with respect to a natural stacking operation. We show that $\Gamma_{n}$ is always a diagram group, in the sense of Guba and Sapir. A number of consequences follow, including biautomaticity and bi-orderability of the groups $\Gamma_{n}$. Moreover, each group $\Gamma_{n}$ acts properly and cocompactly on a CAT(0) cubical complex. (The current version corrects a typographical error and acknowledges overlap with earlier work of Genevois.)