Word length, Morse theory, and Vietoris-Rips complexes

Abstract

We present a discrete Morse theoretic approach to proving high connectivity or contractibility of a Vietoris-Rips complex using distance to a fixed point as an initial measurement. In particular we focus on the case of a finitely generated group using word length. As a proof-of-concept application we prove that $\mathcal{VR}_2(A_\Gamma)$ is contractible for $A_\Gamma$ a right-angled Artin group on a triangle-free graph $\Gamma$. We also prove an interesting sufficient condition for a group to be finitely presented that only requires checking connectivity of certain finite complexes.