Résumé
Algebr. Geom. Topol. 24 (2024) 4007-4060 Starting with a lattice with an action of$\mathbb{Z}$or$\mathbb{R}$ , we build a Helly graph or an injective metric space. We deduce that the$\ell^\infty$orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group or any FC type Artin group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise$\ell^\infty$metric on any Euclidean building of type$\tilde{A_n}$extended,$\tilde{B_n}$ ,$\tilde{C_n}$or$\tilde{D_n}$is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types$\tilde{A_n}$and$\tilde{C_n}$ , we show that the natural piecewise$\ell^\infty$metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the$K(\pi,1)$conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.