Local CAT(0) for the dual braid complex K_d

Determine whether the dual braid complex K_d equipped with the orthoscheme metric is locally CAT(0) for all integers d ≥ 2, which (as noted) would imply that the d-strand braid group Braid_d is a CAT(0) group.

Background

The paper recalls a conjecture from Brady–McCammond (2010) that the dual braid complex K_d should be locally CAT(0). This property has been established for small values of d (up to 7), but is not known in general. The authors’ constructions link polynomial spaces and the dual braid complex, motivating renewed attention to this curvature question.

Establishing local CAT(0) for K_d would have significant group-theoretic consequences, including confirming that Braid_d is a CAT(0) group, thereby connecting geometric combinatorics of polynomials to classical problems in geometric group theory.

References

It was conjectured in that the dual braid complex $K_d$ is locally $(0)$, which would imply that $Braid_d$ is a $(0)$ group. This has been proven for $d \leq 7$ but remains open in general.

— Geometric Combinatorics of Polynomials II: Polynomials and Cell Structures (2410.03047 - Dougherty et al., 4 Oct 2024) in Introduction