Helly meets Garside and Artin

Abstract

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell-Jones conjecture, the coarse Baum-Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a mean of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

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

“…In the case of Garside groups of finite type, we recover a particularly simple proof of the following result by Huang and Osajda (see [HO19]). In particular, our proof does not rely on the deep result that local-to-global result for Helly graphs (Theorem 1.11, see [CCHO21]).…”

“…Note that, in the case of a finite type Garside group, this is due to Huang and Osajda (see [HO19]). However, our proof is different, and does not rely on the deep local-to-global result for Helly graphs (see Theorem 1.11).…”

“…Their discrete counterpart are called Helly graphs. Their use in geometric group theory is recent and growing, see notably [Lan13], [HO19], [CCG + 20], [HHP20], [OV20], [Hae21]. Roughly speaking, CAT(0) spaces are typically locally Euclidean spaces, whereas injective metric spaces are typically locally ℓ ∞ metric spaces.…”

Cubulation of some triangle-free Artin groups

“…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) acts 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 gives rise to classifying spaces with convex geodesic bicombings.…”

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

“…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.…”