{
  "abstract": "We introduce the notion of inner $\\mathrm{C}^{\\ast}$-selfless groups and study their structural properties. We prove a Fourier coefficient extraction theorem for invariant subalgebras of crossed products by such groups under a compatibility assumption on the action. Consequently, every inner $\\mathrm{C}^{\\ast}$-selfless group has the relative $\\mathrm{C}^{\\ast}$-invariant subalgebra rigidity property, and every finite product of such groups has the $\\mathrm{C}^{\\ast}$-invariant subalgebra rigidity property. We also introduce another group-theoretic notion, which we call cyclic Powers property. This implies inner $\\mathrm{C}^{\\ast}$-selflessness, complete $\\mathrm{C}^{\\ast}$-selflessness, and the Powers property, among others. We establish it for groups admitting a topologically free extreme boundary action on a compact Hausdorff space with more than two points, and for Zariski-dense discrete subgroups of $\\mathrm{PSL}(d,\\mathbb K)$ containing a biproximal element, where $\\mathbb K$ is a local field and $d\\geq2$. In particular, we obtain the Powers property for $\\mathrm{PSL}(d,\\mathbb Z)$, $d\\geq3$, answering a question posed more than twenty years ago by Bridson-de la Harpe. Finally, we show that every non-trivial mixed identity-free discrete group has the relative invariant subalgebra rigidity property for its group von Neumann algebra. This shows that the Baumslag-Solitar groups $\\mathrm{BS}(m,n)$ with $|m|,|n|\\geq2$ and $|m|\\ne|n|$ have the relative ISR property, thereby answering a question of Dudko-Jiang.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40198",
  "author_total": 3,
  "authors": [
    {
      "name": "Tattwamasi Amrutam"
    },
    {
      "name": "Mario Klisse"
    },
    {
      "name": "Matteo Pagliero"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vnw9cbke74fj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40198",
  "fulltext_url": null,
  "id": "5vnw9cbke74fj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40198",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40198",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vnw9cbke74fj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40198",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40198"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Inner $\\mathrm{C}^{\\ast}$-Selfless Groups",
  "title_text": "Inner C^∗-Selfless Groups",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5vnw9cbke74fj",
  "venue": "arXiv",
  "versions": []
}