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

TL;DR

This paper presents a simplified proof that the Vietoris-Rips complex of the integer lattice $\

Contribution

It introduces a new, concise proof using Bestvina-Brady Morse theory and establishes a general criterion for contractibility of Vietoris-Rips complexes.

Findings

Provides a shorter proof with improved bounds

Introduces a general criterion for contractibility

Potentially useful for future research in metric space topology

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.