{
  "abstract": "We study the two $\\ell^{2}$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on $\\mathbb{R}^{G}$. Although the two regular representations are unitarily equivalent, their $\\ell^{2}$-Dirichlet subspaces of $\\mathbb{R}^{G}$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $$\\mathcal{D}_{2}\\left(G,λ\\right)=\\mathcal{D}_{2}\\left(G,ρ\\right)\\quad\\Longleftrightarrow\\quad G \\text{ is an FC-group}.$$ The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\\mathbb{Z}$. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.12360",
  "author_total": 2,
  "authors": [
    {
      "name": "Nachi Avraham-Re'em"
    },
    {
      "name": "Zemer Kosloff"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5kapq44kk30w3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.12360",
  "fulltext_url": null,
  "id": "5kapq44kk30w3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.12360",
  "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.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7161054197",
  "pdf_url": "https://arxiv.org/pdf/2605.12360",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5kapq44kk30w3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2605.12360",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.12360"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Asymmetry of $\\ell^{2}$-cohomology via skewed Følner geometry",
  "title_text": "Asymmetry of ℓ^2-cohomology via skewed Følner geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/5kapq44kk30w3",
  "venue": "arXiv",
  "versions": []
}