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Peer-reviewedCC BY 4.0Forum of Mathematics, Sigma

Selberg, Ihara and Berkovich

  • Jialun Li1
  • Carlos Matheus2
  • Wenyu Pan3
  • Zhongkai Tao4
4 affiliations
  1. 1Fudan University
  2. 2CNRS-Centre de Mathématiques Laurent Schwartz, École Polytechnique
  3. 3Department of Mathematics, University of Toronto
  4. 4Institut des Hautes Études Scientifiques
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Abstract

We use the Selberg zeta function to analyze the limiting behavior of resonances in a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated with the relevant non-Archimedean Schottky group acting on the Berkovich projective line.

To reveal more interesting and intricate structures of resonances, we introduce intermediate zeta functions that encapsulate both Archimedean and non-Archimedean information – whereas the classical Selberg and Ihara zeta functions reflect only Archimedean and non-Archimedean properties, respectively. As an application of our techniques, we derive an exponential error term in a result of McMullen (recently extended by Dang–Mehmeti) concerning the asymptotic vanishing rate of the Hausdorff dimension of limit sets of certain degenerating Schottky groups generating symmetric three-funnel surfaces.

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Cite

BibTeXRIS
BibTeX
@article{li2026selberg,
  title = {{Selberg, Ihara and Berkovich}},
  author = {Jialun Li and Carlos Matheus and Wenyu Pan and Zhongkai Tao},
  journal = {Forum of Mathematics, Sigma},
  year = {2026},
  month = nov,
  doi = {10.1017/fms.2026.10279},
  url = {https://www.cambridge.org/core/product/identifier/S2050509426102795/type/journal_article},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Selberg, Ihara and Berkovich
AU  - Jialun Li
AU  - Carlos Matheus
AU  - Wenyu Pan
AU  - Zhongkai Tao
PY  - 2026
DA  - 2026/11/03
JO  - Forum of Mathematics, Sigma
DO  - 10.1017/fms.2026.10279
UR  - https://www.cambridge.org/core/product/identifier/S2050509426102795/type/journal_article
AB  - We use the Selberg zeta function to analyze the limiting behavior of resonances in a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated with the relevant non-Archimedean Schottky group acting on the Berkovich projective line.  To reveal more interesting and intricate structures of resonances, we introduce intermediate zeta functions that encapsulate both Archimedean and non-Archimedean information – whereas the classical Selberg and Ihara zeta functions reflect only Archimedean and non-Archimedean properties, respectively. As an application of our techniques, we derive an exponential error term in a result of McMullen (recently extended by Dang–Mehmeti) concerning the asymptotic vanishing rate of the Hausdorff dimension of limit sets of certain degenerating Schottky groups generating symmetric three-funnel surfaces.
SN  - 2050-5094
ER  -