Bestvina–Brady discrete Morse theory and Vietoris–Rips complexes
Bestvina–Brady discrete Morse theory and Vietoris–Rips complexes
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Bestvina–Brady 离散莫尔斯理论和 Vietoris–Rips 复合体
DOI：
10.1353/ajm.2022.0026
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发表时间：
2018
影响因子：
1.7
通讯作者：
M. C. B. Zaremsky
中科院分区：
文献类型：
--
作者：
M. C. B. Zaremsky
利用Bestoris--布雷迪离散莫尔斯理论的一个新的推广，研究了某些度量空间$X$的Vietoris-Rips复形${\cal{VR}}_t（X）$。我们的主要结果是$X$上的一对度量准则，称为{\it莫尔斯准则}和{\it Link准则}，这使我们能够推导出关于某些${\cal{VR}}_t（X）$的同伦类型的信息。一个应用是拓扑数据分析，特别是持久性同伦类型的某些Vietoris-RIP复合物。例如，我们恢复了Adamaszek-亚当斯和豪斯曼关于${\cal{VR}}_t（S^n）$的同伦类型的一些结果.另一个应用是几何群论，我们证明了任何一组几何上的度量空间满足一个版本的链接标准承认几何行动的可收缩单纯复形，这对有限性的影响组。例如，这适用于渐近${\rm CAT}（0）$群。我们还证明了，任何一个字度量满足链接准则在适当的范围内的组有一个可收缩的Vietoris--Rips复杂，并使用combings表现出家庭的群体与此属性。
abstract:We inspect Vietoris--Rips complexes ${\cal{VR}}_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina--Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the {\it Morse Criterion} and {\it Link Criterion}, that allow us to deduce information about the homotopy types of certain ${\cal{VR}}_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris--Rips complexes. For example we recover some results of Adamaszek--Adams and Hausmann regarding homotopy types of ${\cal{VR}}_t(S^n)$. Another application is to geometric group theory; we prove that any group acting geometrically on a metric space satisfying a version of the Link Criterion admits a geometric action on a contractible simplicial complex, which has implications for the finiteness properties of the group. This applies for example to asymptotically ${\rm CAT}(0)$ groups. We also prove that any group with a word metric satisfying the Link Criterion in an appropriate range has a contractible Vietoris--Rips complex, and use combings to exhibit a family of groups with this property.