{
  "abstract": "We construct two word metrics on the same seven-dimensional torsion-free nilpotent lattice whose asymptotic cones are isometric, but for which every quasi-isometry has multiplicative constant at least $1+\\varepsilon$, for one fixed $\\varepsilon>0$. We use a convex invariant of a fixed quasi-isometry $Ψ:N\\to M$. The displacement cocycle of $Ψ$ and the right-invariant means on $N$ determine a compact convex set $\\mathcal A(Ψ)\\subseteq\\operatorname{Hom}(V_N,V_M)$. If $\\mathscr G_{\\mathrm{pt}}(Ψ)$ denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of $Ψ$ with varying basepoints, then $$ \\mathcal A(Ψ)=\\operatorname{conv}\\mathscr G_{\\mathrm{pt}}(Ψ), \\qquad\\operatorname{Ext}\\mathcal A(Ψ)\\subseteq\\mathscr G_{\\mathrm{pt}}(Ψ). $$ For every extreme point $L\\in\\mathcal A(Ψ)$, the dual map $L^*$ is also the degree-one part of an isomorphism of the full real cohomology rings. This cohomological compatibility, together with the realization of first-layer Pansu differentials by invariant means, yields the multiplicative constant gap stated above.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13055",
  "author_total": 3,
  "authors": [
    {
      "name": "Mayukh Mukherjee"
    },
    {
      "name": "Soumyadeb Samanta"
    },
    {
      "name": "Soumyadip Thandar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8r0msagwcnv3t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13055",
  "fulltext_url": null,
  "id": "8r0msagwcnv3t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13055",
  "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.DG"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221524830",
  "pdf_url": "https://arxiv.org/pdf/2610.13055",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8r0msagwcnv3t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13055",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13055"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups",
  "title_text": "Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/8r0msagwcnv3t",
  "venue": "arXiv",
  "versions": []
}