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

Abstract

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} ${\mathrm{Cross}}^{△}(X)$ 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 ${\mathrm{Cross}}^{△}(X)$ is homotopy equivalent to the pointed sum of the links of all the vertices in the prism-completion $X^{\square }$ of $X$.

43 pages, 12 figures. Comments are welcome!