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Spectral gaps of double covers of hyperbolic surfaces

  • Lawford Hatcher
  • Bram Petri
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Abstract

We prove that, in most moduli spaces of orientable hyperbolic surfaces of finite area, there exist surfaces all of whose covers of degree two have a new small eigenvalue. On the other hand, we show that four punctured spheres always admit a double cover without new small eigenvalues and that there exist closed surfaces of any genus with the same property. We prove the last two of these results using a version of Cheeger's inequality adapted to double covers.

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Cite

BibTeXRIS
BibTeX
@misc{hatcher2026spectral,
  title = {{Spectral gaps of double covers of hyperbolic surfaces}},
  author = {Lawford Hatcher and Bram Petri},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07193},
  eprint = {2610.07193},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07193},
}
RIS
TY  - UNPB
TI  - Spectral gaps of double covers of hyperbolic surfaces
AU  - Lawford Hatcher
AU  - Bram Petri
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07193
UR  - https://arxiv.org/abs/2610.07193
AB  - We prove that, in most moduli spaces of orientable hyperbolic surfaces of finite area, there exist surfaces all of whose covers of degree two have a new small eigenvalue. On the other hand, we show that four punctured spheres always admit a double cover without new small eigenvalues and that there exist closed surfaces of any genus with the same property. We prove the last two of these results using a version of Cheeger's inequality adapted to double covers.
ER  -