Lattices, Garside structures and weakly modular graphs

Abstract

In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural A˜n shaped simplices, including Euclidean buildings of type A˜n and Cayley graphs of Garside groups and their quotients by the Garside elements. All these examples fit into the more general setting of lattices with order-increasing Z-actions and the associated lattice quotients proposed in a previous work by the first named author. We show that both the lattice quotients and the lattices themselves give rise to weakly modular graphs, which is a form of combinatorial non-positive curvature. We also show that several other objects fit into this setting of lattices/lattice quotients, including Artin complexes of Artin-Tits groups of type A˜n, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Hence our result also implies to these objects and shows that they give weakly modular graphs. Along the way, we also clarify the relationship between categorical Garside structures, lattices with Z action and different classes of complexes studied this article. We use this point of view to describe the first examples of Garside groups with exotic properties, like non-linearity or rigidity results.