The 6-strand braid group is CAT(0)

TL;DR

This paper proves that braid groups with up to 6 strands are CAT(0) spaces and establishes that orthoscheme complexes of certain lattices are also CAT(0), advancing understanding of geometric group properties.

Contribution

It demonstrates the CAT(0) property for braid groups with up to 6 strands and for orthoscheme complexes of bounded graded modular complemented lattices, linking group theory and geometric structures.

Findings

Braid groups with ≤6 strands are CAT(0).

Orthoscheme complexes of certain lattices are CAT(0).

Provides partial proof related to Brady and McCammond's conjecture.

Abstract

We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.