Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lattices, injective metrics and the $K(\pi,1)$ conjecture   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 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.

11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th