Vietoris-Rips complexes of affine Coxeter groups
Abstract
We prove that all affine Coxeter groups admit contractible Vietoris-Rips complexes. More precisely, for any affine Coxeter group Wa there exists a finite generating set A such that 𝒱ℛt(Wa;A) is contractible for all t≥ 1. We prove this as a special case of a more general result about certain groups. Let G be a group and H a subgroup. A subgroup F≤ G is a supplement of H in G if G=HF. We prove that for H≤ G with F a supplement of H in G, for generating sets AH and AG of H and G satisfying certain conditions involving F, the Vietoris-Rips complexes 𝒱ℛt(G;AG) and 𝒱ℛt(H;AH) are homotopy equivalent for all t≥ 1. In particular this applies to affine Coxeter groups, since free abelian groups admit contractible Vietoris-Rips complexes. Beyond these general results, we also investigate affine Coxeter groups Wa specifically using the standard generating set S, and prove contractibility of 𝒱ℛt(Wa;S) for large enough t.
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BibTeX
@misc{decker2026vietorisrips,
title = {{Vietoris-Rips complexes of affine Coxeter groups}},
author = {Riley Decker and Matthew C. B. Zaremsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38703},
eprint = {2610.38703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38703},
}
RIS
TY - UNPB
TI - Vietoris-Rips complexes of affine Coxeter groups
AU - Riley Decker
AU - Matthew C. B. Zaremsky
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38703
UR - https://arxiv.org/abs/2610.38703
AB - We prove that all affine Coxeter groups admit contractible Vietoris-Rips complexes. More precisely, for any affine Coxeter group W_a there exists a finite generating set A such that 𝒱ℛ_t(W_a;A) is contractible for all t≥ 1. We prove this as a special case of a more general result about certain groups. Let G be a group and H a subgroup. A subgroup F≤ G is a supplement of H in G if G=HF. We prove that for H≤ G with F a supplement of H in G, for generating sets A_H and A_G of H and G satisfying certain conditions involving F, the Vietoris-Rips complexes 𝒱ℛ_t(G;A_G) and 𝒱ℛ_t(H;A_H) are homotopy equivalent for all t≥ 1. In particular this applies to affine Coxeter groups, since free abelian groups admit contractible Vietoris-Rips complexes. Beyond these general results, we also investigate affine Coxeter groups W_a specifically using the standard generating set S, and prove contractibility of 𝒱ℛ_t(W_a;S) for large enough t.
ER -