From Trees to Tripods: Proof of $K(\pi,1)$ for Artin groups with $ABI$-type spherical parabolics

TL;DR

This paper proves the $K(\pi,1)$-conjecture for a broad class of Artin groups by reducing the problem to tripod-shaped Coxeter diagrams and utilizing properties of injective metric spaces and bi-Helly graphs.

Contribution

It introduces a reduction theorem linking tree and tripod Coxeter diagrams and applies it to establish the conjecture for many Artin groups with specific parabolic subgroup restrictions.

Findings

Proves the $K(\pi,1)$-conjecture for Artin groups with certain spherical parabolics.

Develops a new reduction theorem based on injective metric space towers.

Introduces combinatorial convexity and a Bestvina-type inequality in injective orthoscheme complexes.

Abstract

We reduce the $K(\pi,1)$-conjecture for all Artin groups with tree Coxeter diagrams to properties of Artin groups with tripod-shaped Coxeter diagrams. Combining this reduction theorem and properties of braid groups in previous works of Charney, Crisp-McCammond, Haettel and the second named author, we deduce that the $K(\pi,1)$-conjecture holds for every Artin group whose spherical parabolic subgroups avoid type $D_n$ ($n \ge 4$) and the exceptional types. The reduction theorem relies on producing a ``tower'' of injective metric spaces from a single Artin group. The construction of such a tower relies on two ingredients of independent interests: a notion of combinatorial convexity and a Bestvina-type inequality, in certain injective orthoscheme complexes. These ingredients further rely on the use of structural properties of bi-Helly graphs (also known as absolute bipartite retracts) developed in joint work of the first named author with Munro.