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 Γ ▷◁ 1 and N Γ ▷◁ 2 are homotopy equivalent, where Γ ▷◁ i 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.

2010 Mathematics subject classification. Primary 20F65