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 a 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 means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

The Farrell-Jones Conjecture for normally poly-free groups

“…Conjecture for all Artin groups of FC-type [13]. Sayed K. Roushon has also proved the conjecture for some classes of Artin groups [19].…”

Locally elliptic actions, torsion groups, and nonpositively curved spaces

“…This also generalizes a theorem of Gromov for translation lengths of hyperbolic elements in a Gromov-hyperbolic group (see [Gro87,8.5.S]). Since Garside groups are Helly according to [HO21], this implies a direct analogue of [LL07] for a very closely related translation length. This has consequences in particular for decision problems, following [LL07].…”

“…Lang showed that the any Gromov hyperbolic group acts properly cocompactly on the Helly hull of any Cayley graph (see [Lan13, CCG + 20]). Huang and Osajda proved that any weak Garside group and any Artin group of type FC has a proper and cocompact action on a Helly graph (see [HO21]). Osajda and Valiunas proved that any group that is hyperbolic relative to Helly groups is Helly (see [OV20]).…”

Characterizations of higher rank hyperbolicity

“…The cone inequalities (CI n ) hold in particular if X is a CAT(0) space or a space with a conical geodesic bicombing [10,31]. Every hyperbolic group acts geometrically on a proper polyhedral complex with such a structure [26] (see also [7,21] for more groups with this property).…”