BESTVINA–BRADY DISCRETE MORSE THEORY AND VIETORIS–RIPS COMPLEXES

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 (Sⁿ ). 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.