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 $Γ_1$ and $Γ_2$, if the two right-angled Artin groups $A(Γ_1)$ and $A(Γ_2)$ are quasi-isometric, then the infinite pointed sums $\bigvee_\mathbb{N} Γ_1^{\bowtie}$ and $\bigvee_\mathbb{N} Γ_2^{\bowtie}$ are homotopy equivalent, where $Γ_i^{\bowtie}$ denotes the simplicial complex whose vertex-set is $Γ_i$ 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 \emph{crossing complex} $\mathrm{Cross}^\triangle(X)$ 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 $\mathrm{Cross}^\triangle(X)$ is homotopy equivalent to the pointed sum of the links of all the vertices in the prism-completion $X^\square$ of $X$.