{
  "abstract": "Our results imply, in particular, that a finitely generated solvable group $G$ is nilpotent if and only if it contains a solution to any unimodular equation, i.e., an equation of the form $\\prod g_ix^{n_i}=1$, where $g_i\\in G$ and $\\sum n_i=\\pm1$. A similar fact turns out to be true for Lie algebras. We also exhibit an example of a unimodular equation $w(x)=g$ over a finitely generated group $G$, which has a solution (in $G$) for any $g\\in G$, but the solution is not unique for some $g\\in G$. We show that, for nilpotent groups $G$, the set of unimodular mappings $G^n\\to G^n$ (which are defined naturally) forms a group under the composition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28045",
  "author_total": 3,
  "authors": [
    {
      "name": "Anton A. Klyachko"
    },
    {
      "name": "Mikhail A. Mikheenko"
    },
    {
      "name": "Alexander Yu. Olshanskii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40dkac6qft9dq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28045",
  "fulltext_url": null,
  "id": "40dkac6qft9dq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28045",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7204760820",
  "pdf_url": "https://arxiv.org/pdf/2608.28045",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40dkac6qft9dq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2608.28045",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28045"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Solvability of unimodular equations in groups and Lie algebras",
  "title_text": "Solvability of unimodular equations in groups and Lie algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/40dkac6qft9dq",
  "venue": "arXiv",
  "versions": []
}