{
  "abstract": "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 $\\mathcal{VR}_t(W_a;A)$ is contractible for all $t\\ge 1$. 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 $F\\le G$ is a supplement of $H$ in $G$ if $G=HF$. We prove that for $H\\le G$ 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 $\\mathcal{VR}_t(G;A_G)$ and $\\mathcal{VR}_t(H;A_H)$ are homotopy equivalent for all $t\\ge 1$. 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 $\\mathcal{VR}_t(W_a;S)$ for large enough $t$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38703",
  "author_total": 2,
  "authors": [
    {
      "name": "Riley Decker"
    },
    {
      "name": "Matthew C. B. Zaremsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8cmmwjmgrbpcd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38703",
  "fulltext_url": null,
  "id": "8cmmwjmgrbpcd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38703",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8cmmwjmgrbpcd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Vietoris-Rips complexes of affine Coxeter groups",
  "title_text": "Vietoris-Rips complexes of affine Coxeter groups",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/8cmmwjmgrbpcd",
  "venue": "arXiv",
  "versions": []
}