{
  "abstract": "We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II$_1$ factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras $L(\\operatorname{SL}_3(\\mathbb Z))$ and $L \\mathbb F_2$ are not elementarily equivalent, and we show that the group von Neumann algebra $L\\mathbb F_2$ is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II$_1$ factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which can be taken to be group von Neumann algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.16669",
  "author_total": 1,
  "authors": [
    {
      "name": "Jesse Peterson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2e86ad89qck1a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.16669",
  "fulltext_url": null,
  "id": "2e86ad89qck1a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.16669",
  "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"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7161730059",
  "pdf_url": "https://arxiv.org/pdf/2605.16669",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2e86ad89qck1a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2605.16669",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.16669"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On ultraproduct approximations and property (T) factors",
  "title_text": "On ultraproduct approximations and property (T) factors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/2e86ad89qck1a",
  "venue": "arXiv",
  "versions": []
}