{
  "abstract": "For every integer $n\\ge 2$, we introduce a new combinatorial condition PP$(n)$ inspired by paradoxical decompositions of groups. As $n$ increases, the property PP$(n)$ weakens, and the resulting hierarchy interpolates between group-theoretic manifestations of negative curvature and proper proximality. More precisely, we prove that a countable group is properly proximal if and only if it satisfies PP$(n)$ for some $n$. On the other hand, every acylindrically hyperbolic group satisfies PP$(2)$; furthermore, finitely generated PP$(2)$-groups admit a geometric characterization: they are precisely the groups containing strongly quasi-convex, non-cyclic, free subgroups. In particular, we obtain that every countable acylindrically hyperbolic group is properly proximal. Finally, for every $n\\in\\mathbb N$, we provide examples of finitely generated properly proximal groups that do not satisfy PP$(n)$, thus proving that our hierarchy is indeed infinite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21896",
  "author_total": 3,
  "authors": [
    {
      "name": "D. Osin"
    },
    {
      "name": "K. Toyosawa"
    },
    {
      "name": "Z. Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9gqjpx225kwaf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21896",
  "fulltext_url": null,
  "id": "9gqjpx225kwaf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21896",
  "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.OA"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7213924940",
  "pdf_url": "https://arxiv.org/pdf/2609.21896",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9gqjpx225kwaf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21896",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21896"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A paradoxical route from hyperbolic geometry to proper proximality",
  "title_text": "A paradoxical route from hyperbolic geometry to proper proximality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/9gqjpx225kwaf",
  "venue": "arXiv",
  "versions": []
}