{
  "abstract": "Let $G$ be a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture. Assuming that $b_1^{(2)}(G)=0$, we associate to each surjective integral character $ϕ$ the quantity $b_1^{(2)}(\\ker ϕ)$. We prove that the resulting function extends to a seminorm on $H^1(G;\\mathbb{R})$ and is the thickness function of a polytope in $H_1(G;\\mathbb{R})$. More generally, assuming that $G$ is of type $\\mathrm{FP}_k(\\mathbb{Q})$ and has $b_k^{(2)}(G)=0$, we prove that the function assigning $b_k^{(2)}(\\ker ϕ)$ to each surjective integral character $ϕ$ is likewise the thickness function of a polytope. This confirms a conjecture of Friedl, Lück, and Tillmann. For torsion-free virtually (finitely generated free)-by-cyclic groups, we relate the resulting invariant to the $L^2$-Betti complexity of a character, thereby confirming a conjecture of Gardam and Kielak. As an application, we establish the existence of an algorithm for computing the Bieri--Neumann--Strebel invariant of free-by-cyclic groups and discuss connections with the isomorphism problem for such groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.31774",
  "author_total": 3,
  "authors": [
    {
      "name": "Andrei Jaikin-Zapirain"
    },
    {
      "name": "Monika Kudlinska"
    },
    {
      "name": "Pablo Sánchez-Peralta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/643w0jw7vjrhz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.31774",
  "fulltext_url": null,
  "id": "643w0jw7vjrhz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.31774",
  "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.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7166820730",
  "pdf_url": "https://arxiv.org/pdf/2606.31774",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/643w0jw7vjrhz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2606.31774",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.31774"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Thurston norm, polytopes and splitting complexity",
  "title_text": "Thurston norm, polytopes and splitting complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/643w0jw7vjrhz",
  "venue": "arXiv",
  "versions": []
}