  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.
