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 FarrellâJones conjecture, the coarse BaumâConnes 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.
Highlights
We prove that the universal covers of the Salvetti complexes of higher dimensional Artin groups of type FC are cell Helly
We present an alternative proof of the fact that the Salvetti complex for Artin groups of type FC is aspherical, which implies the K (Ï, 1)-conjecture for such groups
One important feature of injective metric spaces of finite dimension is that they admit a unique convex, consistent, reversible geodesic bicombing [44, Theorems 1.1&1.2]
Summary
In the Corollary below we list immediate consequences of being Helly, for groups as in the Main Theorem. One important feature of injective metric spaces of finite dimension is that they admit a unique convex, consistent, reversible geodesic bicombing [44, Theorems 1.1&1.2]. (2) to a CAT(0) space, an injective metric space (or, more generally, a space with a convex geodesic bicombing) has a boundary at infinity, with two different topologies: the cone topology and the Tits topology [44,66]. The former gives rise to an EZ-boundary (cf [15,49]) of G [44]. Weak Garside groups of finite type are biautomatic by [35,41]
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