Contractible Vietoris-Rips complexes of ${\mathbb{Z}}^n$

Abstract

We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group ${\mathbb{Z}}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we use Bestvina-Brady discrete Morse theory to get a very short proof with better bounds. In the course of this, we get a new, general criterion for a metric space to have contractible Vietoris-Rips complexes at large enough scales, which could prove useful in the future.

arXiv comment5 pages. V2: Fixed some errors, incorporated referee suggestions; accepted version, to appear in Proc. Amer. Math. Soc