Lattices, Garside structures and weakly modular graphs

TL;DR

This paper explores the combinatorial non-positive curvature properties of various complexes related to lattices, Garside groups, and Artin groups, establishing their weakly modular graph structures and connecting different algebraic and geometric frameworks.

Contribution

It demonstrates that lattice quotients and lattices themselves form weakly modular graphs, unifying several complexes under a common non-positive curvature framework and clarifying their algebraic-geometric relationships.

Findings

Lattice quotients and lattices are weakly modular graphs.

Artin complexes of type f0A_n are included in this framework.

Garside groups can exhibit exotic properties like non-linearity.

Abstract

In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural $\widetilde A_n$ shaped simplicies, including Euclidean buildings of type $\widetilde A_n$ 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 $\mathbb Z$-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 complexes fit into this setting of lattices/lattice quotients, hence our result applies, including Artin complexes of Artin-Tits groups of type $\widetilde A_n$, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Along the way, we also clarify the relationship between categorical Garside structure, lattices with $\mathbb Z$ 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.