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

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arxiv 2410.11993 v2 pith:43KWV477 submitted 2024-10-15 math.GR math.GTmath.MG

Matthew C. B. Zaremsky This is my paper · ORCID
classification math.GRmath.GTmath.MG
keywords contractibleproofvietoris-ripscomplexesenoughlargemathbbmetric
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture

    math.GT 2025-04 accept novelty 8.0 of 10

    Colorable hierarchically hyperbolic groups admit asymptotically CAT(0) metrics and Bestvina-Dranishnikov Z-structures, leading to the Farrell-Jones conjecture for extra-large type Artin groups and other new classes.

  2. Vietoris-Rips complexes of torus grids

    math.AT 2025-02 conditional novelty 7.0 of 10

    Vietoris-Rips complexes of n-by-n torus grids are shown to be tori, spheres, or wedges of spheres for several infinite families of grid sizes and scales.

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