Lattices, injective metrics and the $K(\pi ,1)$ conjecture

Abstract

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 $\widetilde{A_n}$ extended, $\widetilde{B_n}$, $\widetilde{C_n}$ or $\widetilde{D_n}$ is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types $\widetilde{A_n}$ and $\widetilde{C_n}$, 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.

47 pages, 4 figures. v5: typos corrected, final accepted version

References in corpus (1)

• The conjugacy stability problem for parabolic subgroups in Artin groups