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

TL;DR

This work advances the quasi-isometry classification of right-angled Artin groups by connecting algebraic quasi-isometries to the homotopy types of hyperplane complexes in quasi-median graphs. It develops a robust toolkit for complexes of hyperplanes, including crossing, contact, and relative variants, and shows how these are governed by prism-completions and wedge decompositions. By translating coset-intersection data into homotopy invariants, the authors derive new quasi-isometry and commensurability invariants for graph products, notably yielding a topological fingerprint $\bigvee_{\mathbb{N}} \Gamma^{\bowtie}$ for the defining graph $\Gamma$. The results have broad implications for large-scale geometry of RAAGs and prompt several open questions about extending invariants to broader graph-product families and to Coxeter groups.

Abstract

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 $X$.

Figures (10)

• Figure 1: A quasi-median graph and some of its hyperplanes. The orange hyperplane is transverse to the red, blue, and purple hyperplanes. The purple hyperplane is in contact with the red, blue, and orange hyperplanes. The orange and green hyperplanes are in contact. The red and blue hyperplanes are not in contact.
• Figure 2: A quasi-median graph, and its contact and crossing complexes.
• Figure 3: A quasi-median graph $X$ endowed with a collection of gated subgraphs $\mathbb{G}$ and the corresponding $\mathbb{G}$-contact complex.
• Figure 4: A quasi-median graph and some of its hyperplanes. The orange hyperplane is transverse to the red, blue, and purple hyperplanes. The green and orange hyperplanes are tangent but not contiguous. The purple hyperplane is contiguous to the red and blue hyperplanes. The red and blue hyperplanes are neither transverse, nor tangent, nor contiguous.
• Figure 5: A quasi-median graph, and its contact and crossing complexes.

Theorems & Definitions (99)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2
• Proposition 2.3: QM
• Lemma 2.4
• Theorem 2.5
• Corollary 2.6
• proof