The planar pure braid group is a diagram group

2021/09/07 by Farley, Daniel S.
#20F36 (57M07) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2109.02815

Abstract

A planar pure braid consists of n descending smooth arcs, each connecting a point on one horizontal line ℓ₁ to a point on a horizontal line ℓ₂, which is required to be directly below the first point. Two arcs are allowed to cross, but no threefold intersections are allowed. The set Γ_n of all planar pure braids on n strands is a group with respect to a natural stacking operation. We show that Γ_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 Γ_n. Moreover, each group Γ_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.)

Related

• A presentation for the planar pure braid group
• Braids in trivial braid diagrams
• On Marked Braid Groups
• Diagrammatics for Coxeter groups and their braid groups
• Braids, trees, and operads