{
  "abstract": "We introduce a new invariant associated to an action of a group $G$ on a (possibly non-commutative) group $K$. The invariant is based on the $G$-parametrized LS-category of the semi-direct product $K\\rtimes G$ but it also depends on a choice of a derivation $d\\colon G\\to K$. The invariant is thus denoted $\\operatorname{tc}(d)$ and it takes integer values between $\\operatorname{cd}(K)$ and $\\operatorname{cd}(K)+\\operatorname{cd}(G)$ (where $\\operatorname{cd}(G)$ stands for the cohomological dimension of a group). We develop a general theory and then compute its values for a number of important examples, including fundamental groups of mapping tori, of pure braid groups and of iterated semi-direct products of free groups. To achieve this, we relate $\\operatorname{tc}(d)$ to the cohomological dimension of its factors, to the sectional category of the inclusion of $G$ in $K\\rtimes G$, and to the properties of torsors that are associated to the derivation $d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01462",
  "author_total": 2,
  "authors": [
    {
      "name": "Urban Ogrinec"
    },
    {
      "name": "Petar Pavešić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajwj5rb56qgdd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01462",
  "fulltext_url": null,
  "id": "ajwj5rb56qgdd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01462",
  "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.AT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7206213066",
  "pdf_url": "https://arxiv.org/pdf/2609.01462",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajwj5rb56qgdd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01462",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01462"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Parametrized LS category and group actions",
  "title_text": "Parametrized LS category and group actions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/ajwj5rb56qgdd",
  "venue": "arXiv",
  "versions": []
}