On the spectral gap of negatively curved surface covers
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909582900592640 |
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| author | Hide, Will Moy, Julien Naud, Frederic |
| author_facet | Hide, Will Moy, Julien Naud, Frederic |
| contents | Given a negatively curved compact Riemannian surface $X$, we give an explicit estimate, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of $X$. The explicit gap is given in terms of the bottom of the spectrum of the universal cover of $X$ and the topological entropy of the geodesic flow on X. This result generalizes in variable curvature a result of Magee-Naud-Puder for hyperbolic surfaces. We then formulate a conjecture on the optimal spectral gap and show that there exists covers with near optimal spectral gaps using a result of Louder-Magee and techniques of strong convergence from random matrix theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10733 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the spectral gap of negatively curved surface covers Hide, Will Moy, Julien Naud, Frederic Spectral Theory 35P15, 37D40, 60B20 Given a negatively curved compact Riemannian surface $X$, we give an explicit estimate, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of $X$. The explicit gap is given in terms of the bottom of the spectrum of the universal cover of $X$ and the topological entropy of the geodesic flow on X. This result generalizes in variable curvature a result of Magee-Naud-Puder for hyperbolic surfaces. We then formulate a conjecture on the optimal spectral gap and show that there exists covers with near optimal spectral gaps using a result of Louder-Magee and techniques of strong convergence from random matrix theory. |
| title | On the spectral gap of negatively curved surface covers |
| topic | Spectral Theory 35P15, 37D40, 60B20 |
| url | https://arxiv.org/abs/2502.10733 |