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In this article we study combinatorial non-positive curvature aspects ofvarious simplicial complexes with natural $\widetilde A_n$ shaped simplicies,including Euclidean buildings of type $\widetilde A_n$ and Cayley graphs ofGarside groups and their quotients by the Garside elements. All these examplesfit into the more general setting of lattices with order-increasing $\mathbbZ$-actions and the associated lattice quotients proposed in a previous work bythe first named author. We show that both the lattice quotients and thelattices themselves give rise to weakly modular graphs, which is a form ofcombinatorial non-positive curvature. We also show that several other complexesfit into this setting of lattices/lattice quotients, hence our result applies,including Artin complexes of Artin-Tits groups of type $\widetilde A_n$, aclass of arc complexes and weak Garside groups arising from a categoricalGarside structure in the sense of Bessis. Along the way, we also clarify therelationship between categorical Garside structure, lattices with $\mathbb Z$action and different classes of complexes studied this article. We use thispoint of view to describe the first examples of Garside groups with exoticproperties, like non-linearity or rigidity results.

Lattices, Garside structures and weakly modular graphs