English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Starting with a lattice with an action of $\mathbb{Z}$ or $\mathbb{R}$, webuild 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. Weapply this to show that any Garside group or any FC type Artin group acts on aninjective metric space and on a Helly graph. We also deduce that the naturalpiecewise $\ell^\infty$ metric on any Euclidean building of type $\tilde{A_n}$extended, $\tilde{B_n}$, $\tilde{C_n}$ or $\tilde{D_n}$ is injective, and itsthickening 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 aHelly 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 whenthe Deligne complex has a CAT(0) metric.

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