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 ${\mathcal{V}\mathcal{R}}_2(A_{\Gamma })$ 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.

arXiv comment14 pages, 3 figures