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., 2024) in Introduction

However, the question whether $X_n$, endowed with the $\ell2$ orthoscheme metric, could be CAT(0) is still open (see). In particular, the question whether braid groups are CAT(0) is still open, but we do know that they are Helly.

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Remark following Theorem 13.?, subsection “Application to Garside groups and Artin groups”

In particular, the question whether braid groups are CAT(0) is still open, but we do know that they are Helly.

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Remark following Theorem 13.?, subsection “Application to Garside groups and Artin groups”