{
  "abstract": "For every rational number $α\\geq 2$, we construct a 2-step nilpotent group with conjugator length function $\\operatorname{CL}(n)\\simeq n^α$. We deduce that the nilpotent conjugator length spectrum is dense in $\\{0\\}\\cup[2,\\infty)$, even after restricting to groups of nilpotency class at most $2$. Moreover, we show that, despite being a commensurability invariant of nilpotent groups, conjugator length is not a quasi-isometry invariant: for every $m \\geq 1$, the cocompact lattices in $\\mathsf H(\\mathbb R)^m$ realize exactly the growth types $n^2,\\ldots,n^{m+1}$. Finally, for every real cubic algebraic number $θ$, we construct a 2-step nilpotent group such that $n^3 \\preceq \\operatorname{CL}(n) \\preceq n^{3+\\varepsilon}$ for every $\\varepsilon > 0$, with $n^3 \\prec \\operatorname{CL}(n)$ if and only if at least one real conjugate of $θ$ has unbounded partial quotients. Consequently, determining the conjugator length function for arbitrary 2-step nilpotent groups is at least as hard as settling the bounded-partial-quotient problem for real cubic algebraic numbers with two nonreal conjugates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14190",
  "author_total": 2,
  "authors": [
    {
      "name": "Jonas Deré"
    },
    {
      "name": "Ken Vandermeersch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f3a5n4grnnx47.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14190",
  "fulltext_url": null,
  "id": "f3a5n4grnnx47",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14190",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7213343083",
  "pdf_url": "https://arxiv.org/pdf/2609.14190",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f3a5n4grnnx47.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14190",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14190"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Conjugator length in nilpotent groups",
  "title_text": "Conjugator length in nilpotent groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/f3a5n4grnnx47",
  "venue": "arXiv",
  "versions": []
}