Metric systolicity and two-dimensional Artin groups

Abstract

We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all twodimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity.

XXL type Artin groups are CAT(0) and acylindrically hyperbolic

“…Many consequences of being CAT(0) are already consequences of being systolic, and as such are consequences of Huang and Osajda's result (see [HO17]). For instance, the Novikov conjecture, the fact that centralizers virtually split, the quadratic Dehn function.…”

“…Concerning variations on the notion of nonpositive curvature, Bestvina defined a geometric action of Artin groups of spherical Artin on a simplicial complex with some nonpositive curvature features (see [Bes99]). More recently, Huang and Osajda proved (see [HO17]) that every Artin group of almost large type (a class including all Artin groups of large type) act properly and cocompactly on systolic complexes, which are a combinatorial variation of nonpositive curvature. They also proved (see [HO19]) that every Artin group of type FC acts geometrically on a Helly graph, which give rise to classifying spaces with convex geodesic bicombings.…”

Injective metrics on buildings and symmetric spaces

“…Chalopin et al proved (see [CCG + 20, Corollary 6.2]) that any type-preserving uniform lattice in a Euclidean building of type C n is Helly. Huang and Osajda proved that any Artin group of type FC is Helly (see [HO19]).…”

Acylindrical hyperbolicity for Artin groups of dimension 2

“…In particular, they are known to satisfy all the above conjectures. Artin groups of type FC and of dimension 2 are well understood and satisfy all of the above conjectures as well [11,12,29]).…”