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.
In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a
natural piecewise $ell^infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural piecewise $ell^infty$ metric which is coarsely Helly. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. The only exception is the special linear group: if $n geq 3$ and $mathbb{K}$ is a local field, we show that $operatorname{SL}(n,mathbb{K})$ does not act properly and coboundedly on an injective metric space.
We prove that some classes of triangle-free Artin groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin groups that are properly cubulated but cannot be cocompactly c
ubulated, even virtually. The existence of such a proper action has many interesting consequences for the group, notably the Haagerup property, and the Baum-Connes conjecture with coefficients.
We describe a simple locally CAT(0) classifying space for extra extra large type Artin groups (with all labels at least 5). Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that extra large type
Artin groups are acylindrically hyperbolic. Together with Property RD proved by Ciabonu, Holt and Rees, the CAT(0) property implies the Baum-Connes conjecture for all XXL type Artin groups.
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixe
d point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Mannings property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem nor on bounded cohomology. More generally, we prove that any morphism from a higher rank lattice to a hierarchically hyperbolic group has finite image. In the Appendix, Vincent Guirardel and Camille Horbez deduce rigidity results for morphisms from a higher rank lattice to various outer automorphism groups.
We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-
dimensional type. A particular case is that for $n geq 4$, the $n$-strand braid group is not virtually cocompactly cubulated.
We show that symmetric spaces and thick affine buildings which are not of spherical type $A_1^r$ have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund.
Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median.
