{
  "abstract": "We study the commensurators of free groups and free pro-p groups, as well as certain subgroups of these. We prove that the commensurator Comm(F ) of a non-abelian free group of finite rank F is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of Comm(F ) and show that some groups in this family are simple. For a prime p, we also consider the p-commensurator Commp(F ), which is the commensurator of F viewed as a group with pro-p topology. By contrast with Comm(F ), we prove that Commp(F ) has a simple subgroup of index at most 2. Further, while the isomorphism class of Comm(F ) does not depend on the rank of F , we prove that the isomorphism class of Commp(F ) depends on the rank of F and determine the exact dependency. If F is the pro-p completion of F (which is a free pro-p group), Comm(F) is a totally disconnected locally compact (tdlc) group containing F as an open subgroup. We use Commp(F ) to construct an abstractly simple subgroup of Comm(F) containing F as well as a family of non-discrete tdlc groups which are compactly generated and simple.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2507.04120",
  "author_total": 6,
  "authors": [
    {
      "name": "Yiftach Barnea"
    },
    {
      "name": "Mikhail Ershov"
    },
    {
      "name": "Adrien Le Boudec"
    },
    {
      "name": "Colin D. Reid"
    },
    {
      "name": "Matteo Vannacci"
    },
    {
      "name": "Thomas Weigel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3aqf4b1m4gt2s.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "3aqf4b1m4gt2s",
  "issns": [],
  "landing_url": "https://hal.science/hal-05768271v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05768271/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3aqf4b1m4gt2s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:42Z",
      "native_id": "hal-05768271",
      "source": "hal",
      "url": "https://hal.science/hal-05768271v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "On commensurators of free groups and free pro-p groups",
  "title_text": "On commensurators of free groups and free pro-p groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:42Z",
  "url": "https://research.draconitic.com/paper/3aqf4b1m4gt2s",
  "venue": "HAL",
  "versions": []
}