Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes

TL;DR

This paper introduces new metric criteria using a generalized Bestvina-Brady Morse theory to analyze the homotopy types of Vietoris-Rips complexes, with applications in topological data analysis and geometric group theory.

Contribution

It develops the Morse and Link Criteria to determine homotopy types of Vietoris-Rips complexes and applies these to groups acting on metric spaces, extending understanding in both data analysis and group theory.

Findings

Recovered homotopy types of $VR_t(S^n)$ for spheres.

Proved groups acting on spaces satisfying the Link Criterion admit actions on contractible complexes.

Identified classes of groups with contractible Vietoris-Rips complexes using combings.

Abstract

We inspect Vietoris-Rips complexes $VR_t(X)$ 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 Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. 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 $VR_t(S^n)$. 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 $CAT(0)$ 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.