{
  "abstract": "We study the Diophantine problem in restricted wreath products of groups. Write $G=B\\wr H$, where $B$ is the passive factor and $H$ is the active factor. The main results demonstrate that in a wide class of wreath products $G$, the Diophantine problem is undecidable. First, we show that if both factors are indicable, then $G$ has a retract isomorphic to $\\mathbb{Z}\\wr\\mathbb{Z}$ and therefore has undecidable Diophantine problem. Then, we show that for any groups $B$ and $H$, the Diophantine problems in $B$ and $H$ are polynomial time reducible to the Diophantine problem in $G$. Finally, we combine e-definability with retractions in order to prove that the Diophantine problem in $G$ is undecidable whenever $B$ is an infinite finitely generated abelian group and $H$ is a finitely generated group containing an infinite e-definable finitely generated abelian subgroup $A$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22719",
  "author_total": 2,
  "authors": [
    {
      "name": "Michael Catli"
    },
    {
      "name": "Alexei Myasnikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fm5zas3qw0rsc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22719",
  "fulltext_url": null,
  "id": "fm5zas3qw0rsc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22719",
  "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": "W7222960934",
  "pdf_url": "https://arxiv.org/pdf/2610.22719",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fm5zas3qw0rsc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22719",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22719"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Diophantine Problem in Wreath Products of Groups",
  "title_text": "The Diophantine Problem in Wreath Products of Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/fm5zas3qw0rsc",
  "venue": "arXiv",
  "versions": []
}