{
  "abstract": "We give a general method for proving Kazhdan's property $(T)$ for quotients $G/K_{[c+1]}$, where $G$ is countable, $K$ is a normal subgroup and $K_{[j]}$ denotes its lower central series. The method combines an affine realization of $G/[K,K]$, a contraction argument, and permanence for nilpotent normal subgroups. We apply this method to several classes of groups by means of homological representations. For the Torelli lower-central series quotients $\\operatorname{Mod}(Σ_g)/(\\mathcal{T}_g)_{[c+1]}$ for $g\\ge3$ we show that they have property $(T)$ for every $c\\ge1$. We also prove property $(T)$ for $\\operatorname{Aut}(F_3)/(\\mathrm{IA}_3)_{[c+1]}$ for every $c\\ge1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. Several other classes of examples are discussed as well. We also prove that the Torelli lower-central series quotients for $c\\ge2$ are not residually finite, and thus non-linear.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12196",
  "author_total": 1,
  "authors": [
    {
      "name": "Piotr W. Nowak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fhwrjqwtwsvzw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12196",
  "fulltext_url": null,
  "id": "fhwrjqwtwsvzw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12196",
  "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.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212861504",
  "pdf_url": "https://arxiv.org/pdf/2609.12196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fhwrjqwtwsvzw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.12196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Property $(T)$ and nonlinearity of mapping class group quotients",
  "title_text": "Property (T) and nonlinearity of mapping class group quotients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/fhwrjqwtwsvzw",
  "venue": "arXiv",
  "versions": []
}