{
  "abstract": "We study two obstructions to a finite group $G$ having a generating $d$-tuple whose Cayley graph has spectral gap at least $ε$. Lubotzky and Weiss proved that such a gap forces $\\mathrm{ab}_k(G)$, the largest order of the abelianization of a subgroup of index at most $k$, to be $\\exp(O_ε(dk))$, while de la Harpe, Robertson and Valette proved that it forces $\\mathrm{rep}_k(G)$, the number of irreducible complex representations of degree at most $k$, to be $\\exp(O_ε(dk^2))$. We show that the first bound implies the second, and in fact implies it with $k^2$ replaced by $k\\log k$. For this, we prove that every finite $d$-generated group satisfies the inequality $\\mathrm{rep}_k(G)\\leq\\exp((d-1)k\\log k+O(dk))\\cdot\\mathrm{ab}_k(G)$, regardless of its spectral gap. The leading term $(d-1)k\\log k$ is attained by powers of alternating groups, and it disappears for groups without large alternating sections. The proof reduces to counting the finite primitive subgroups of $\\mathrm{GL}_m(\\mathbf{C})$, and we show that there are at most $\\exp(O(\\log^4m))$ of them up to conjugacy and scalars. Wigderson asked whether $k^2$ in the second bound can be lowered to $k$, so our result answers his question up to a logarithm. We do not know whether the logarithm is actually needed. For powers of alternating groups, the natural candidates, we show that a spectral gap keeps the number of factors a fixed power below the maximum allowed by $d$-generation, and that random generators have spectral gap tending to $0$ already for superpolynomially many factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37399",
  "author_total": 1,
  "authors": [
    {
      "name": "Urban Jezernik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ebfkb6z8xc2m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37399",
  "fulltext_url": null,
  "id": "1ebfkb6z8xc2m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37399",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7224094815",
  "pdf_url": "https://arxiv.org/pdf/2610.37399",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ebfkb6z8xc2m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37399",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37399"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Representation and abelianization growth, particularly along expanding sequences",
  "title_text": "Representation and abelianization growth, particularly along expanding sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/1ebfkb6z8xc2m",
  "venue": "arXiv",
  "versions": []
}