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A graph is Helly if every family of pairwise intersecting combinatorial ballshas a nonempty intersection. We show that weak Garside groups of finite typeand FC-type Artin groups are Helly, that is, they act geometrically on Hellygraphs. In particular, such groups act geometrically on spaces with convexgeodesic bicombing, equipping them with a nonpositive-curvature-like structure.That structure has many properties of a CAT(0) structure and, additionally, ithas a combinatorial flavor implying biautomaticity. As immediate consequenceswe obtain new results for FC-type Artin groups (in particular braid groups andspherical Artin groups) and weak Garside groups, including e.g.\ fundamentalgroups of the complements of complexified finite simplicial arrangements ofhyperplanes, braid groups of well-generated complex reflection groups, andone-relator groups with non-trivial center. Among the results are:biautomaticity, existence of EZ and Tits boundaries, the Farrell-Jonesconjecture, the coarse Baum-Connes conjecture, and a description of higherorder homological and homotopical Dehn functions. As a mean of proving theHelly property we introduce and use the notion of a (generalized) cell Hellycomplex.

Helly meets Garside and Artin