{
  "abstract": "For every group Γ, we define a homomorphism d Γ : Comm(Γ) → Z (F S) from the abstract commensurator Comm(Γ) to the free abelian group Z (F S) with basis the collection FS of isomorphism classes of finite simple groups. We investigate the homomorphism d Γ when Γ is a finitely generated free group F . We explicitly describe the image of d F : Comm(F ) → Z (F S) , which is a free abelian group of infinite rank, and we show that the kernel is the monolith of the group Comm(F ). We deduce in particular that every proper quotient of Comm(F ) is abelian.\n\nWe use this to study the commensurator of a cocompact lattice in the automorphism group of a regular tree, which can be seen as a subgroup of Comm(F ). We show that the image of this group under d F is again a free abelian group of infinite rank, showing in particular this group is not virtually simple.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35323",
  "author_total": 1,
  "authors": [
    {
      "name": "Adrien Le Boudec"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n70y3tyjfc2t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35323",
  "fulltext_url": null,
  "id": "2n70y3tyjfc2t",
  "issns": [],
  "landing_url": "https://hal.science/hal-05768282v1",
  "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": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214715836",
  "pdf_url": "https://hal.science/hal-05768282/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n70y3tyjfc2t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:42Z",
      "native_id": "hal-05768282",
      "source": "hal",
      "url": "https://hal.science/hal-05768282v1"
    },
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35323",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35323"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "FREE ABELIAN QUOTIENTS OF COMMENSURATORS",
  "title_text": "FREE ABELIAN QUOTIENTS OF COMMENSURATORS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/2n70y3tyjfc2t",
  "venue": "HAL",
  "versions": []
}