Contractible Vietoris–Rips complexes of ℤⁿ

We give a new, short proof of a result of Virk, that the Vietoris–Rips complex of the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.