{
  "abstract": "Let $X$ be proper CAT$(-1)$ and let $Γ\\le\\Isom(X)$ be finitely generated and discrete. The sharp structural theorem states that, if $Γ$ is accessible over finite subgroups and $δ_X(Γ)<1$, then $Γ$ is geometrically finite and virtually free, has a finite graph of groups with finite edge groups and virtually cyclic infinite vertex groups, and $\\partialΓ\\toΛ_Γ$ collapses exactly their conjugate two point boundaries. The result is hereditary, and below $1/2$ every finitely generated subgroup is convex-cobounded (\\cref{thm:accessible-main}). Hence non-virtually-free accessible groups have $δ_X(Γ)\\ge1$ (\\cref{cor:accessible-gap}). Consequences cover finitely presented groups, groups with uniformly bounded finite-subgroup orders, characteristic-zero linear groups, and Kleinian groups, also infinite parabolic-free Kleinian groups have finite-index classical Schottky subgroups (\\cref{cor:accessibility-extension,cor:linear-groups,cor:kleinian-classical}). Hence we cover substantial larger class than \\cite{LiuWang2023},\\cite{Hou2001}, also see \\cref{rem:strictness-sharpness}. Finally, we also state consequences for finite JSJ representatives and hierarchies (\\cref{thm:JSJ,thm:hierarchy,cor:hierarchy-dimension}).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00557",
  "author_total": 1,
  "authors": [
    {
      "name": "Yong Hou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d9ardv66ahhar.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00557",
  "fulltext_url": null,
  "id": "d9ardv66ahhar",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00557",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7206163536",
  "pdf_url": "https://arxiv.org/pdf/2609.00557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d9ardv66ahhar.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00557"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Accessible CAT$(-1)$ groups of critical exponent less than one",
  "title_text": "Accessible CAT(−1) groups of critical exponent less than one",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/d9ardv66ahhar",
  "venue": "arXiv",
  "versions": []
}