{
  "abstract": "Let $Σ$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:Σ\\longrightarrow M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\\varGamma=π_1(Σ)$, let $ρ_M:\\varGamma\\longrightarrow \\mathrm{PSL}_2(\\mathbb{C})$ be the holonomy of $M$ and $ρ:\\varGamma\\longrightarrow \\mathrm{PSL}_2(\\mathbb{C})$ any representation in $\\mathrm{PSL}_2(\\mathbb{C})$. We will show that, if $ρ$ is discrete and non-faithful, then \\[ \\|[\\mathrm{Vol}(ρ)]-[\\mathrm{Vol}(ρ_M)]\\|_\\infty\\geq \\boldsymbol{v}_3 \\] holds, where $[\\mathrm{Vol}(ρ)]$ denotes the bounded fundamental class of $ρ$ in the bounded cohomology $H_b^3(\\varGamma,\\mathbb{R})$ of $\\varGamma$ and $\\boldsymbol{v}_3$ is the volume of a regular ideal 3-simplex in $\\mathbb{H}^3$. As an application, we present a rigidity theorem for $ρ_M$ in the set of representations $ρ$ of $\\varGamma$ in $\\mathrm{PSL}_2(\\mathbb{C})$ in terms of $[\\mathrm{Vol}(ρ)]$. The rigidity theorem implies that $\\boldsymbol{v}_3$ is the optimal separating constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06937",
  "author_total": 2,
  "authors": [
    {
      "name": "Yushi Nakano"
    },
    {
      "name": "Teruhiko Soma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fx6fnvxzn3na2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06937",
  "fulltext_url": null,
  "id": "fx6fnvxzn3na2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06937",
  "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.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212085930",
  "pdf_url": "https://arxiv.org/pdf/2609.06937",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fx6fnvxzn3na2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06937",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06937"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Bounded cohomology and optimal separating constant for representations",
  "title_text": "Bounded cohomology and optimal separating constant for representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fx6fnvxzn3na2",
  "venue": "arXiv",
  "versions": []
}