Bestvina–Brady discrete Morse theory and Vietoris–Rips complexes

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.