Title: Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups Show Chinese title

Title: 准中图的超平面复形的同伦类型及其在右角Artin群中的应用

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} $\mathrm{Cross}^\triangle(X)$ 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 $X$. Show Chinese abstract

Abstract: 在本文中，我们证明，给定两个有限连通图$\Gamma_1$和$\Gamma_2$，如果两个直角Artin群$A(\Gamma_1)$和$A(\Gamma_2)$是拟等距的，那么无限带点和$\bigvee_\mathbb{N} \Gamma_1^{\bowtie}$和$\bigvee_\mathbb{N} \Gamma_2^{\bowtie}$是同伦等价的，其中$\Gamma_i^{\bowtie}$表示顶点集为$\Gamma_i$且单形由连接给出的单纯复形。 这些不变量来源于对准中图中若干超平面复形同伦类型的独立研究(例如CAT(0)立方复形的1-骨架)。 例如，给定一个准中图$X$，\emph{交叉复数}$\mathrm{Cross}^\triangle(X)$是一个单纯复形，其顶点是$X$的超平面(或$\theta$-类)，其单形是成对横截的超平面的集合。 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 $X$.