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Equivalence between CAT(0) properties of polynomial space and dual braid complex

Establish that the stratified Euclidean metric on the space _d^{mt} is CAT(0) if and only if the orthoscheme metric on the dual braid complex K_d is locally CAT(0).

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Background

Building on the first conjecture, the authors posit an equivalence between the global CAT(0) property of the polynomial space (_d{mt}) and the local CAT(0) property of the dual braid complex K_d. This would link curvature in polynomial moduli to curvature in the dual Garside geometric model for braids.

Such an equivalence would unify two geometric frameworks and potentially reduce the CAT(0) question for braid groups to a CAT(0) question for polynomial spaces, and vice versa.

References

\begin{conj}\label{conj:cat0-equivalence} The stratified Euclidean metric on the space $_d{mt}$ is $(0)$ if and only if the orthoscheme metric on the dual braid complex $K_d$ is locally $(0)$. \end{conj}

— Geometric Combinatorics of Polynomials II: Polynomials and Cell Structures (2410.03047 - Dougherty et al., 4 Oct 2024) in Generalizations and conjectures; Conjecture (label 'conj:cat0-equivalence')