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 $\Gamma_1$ and $\Gamma_2$, if the two right-angled Artin groups $A(\Gamma_1)$ and $A(\Gamma_2)$ are quasi-isometric, then the infinite pointed sums $\bigvee_\mathbb{N} \Gamma_1^{\bowtie}$ and $\bigvee_\mathbb{N} \Gamma_2^{\bowtie}$ are homotopy equivalent, where $\Gamma_i^{\bowtie}$ denotes the simplicial complex whose vertex-set is $\Gamma_i$ 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 $\theta$-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 $X$.