Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups

In this article, we prove that, given two finite connected graphs Γ₁ and Γ₂, if the two right-angled Artin groups A(Γ₁) and A(Γ₂) are quasi-isometric, then the infinite pointed sums N Γ₁^ and N Γ₂^ are homotopy equivalent, where Γᵢ^ denotes the simplicial complex whose vertex-set is Γᵢ and whose simplices are given by joins. These invariants are extracted from a study, of independent interest, of the homotopy types of several complexes of hyperplanes in quasi-median graphs (such as one-skeleta of CAT(0) cube complexes). For instance, given a quasi-median graph X, the crossing complex Cross⁽X) is the simplicial complex whose vertices are the hyperplanes (or θ-classes) of X and whose simplices are collections of pairwise transverse hyperplanes. When X has no cut-vertex, we show that Cross⁽X) is homotopy equivalent to the pointed sum of the links of all the vertices in the prism-completion X^ of X.