Spectral gap of random covers of negatively curved noncompact surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908360043921408 |
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| author | Moy, Julien |
| author_facet | Moy, Julien |
| contents | Let $(X,g)$ be a complete noncompact geometrically finite surface with pinched negative curvature $-b^2\leq K_g \leq -1$. Let $λ_0(\widetilde{X})$ denote the bottom of the $L^2-$spectrum of the Laplacian on the universal cover $\widetilde{X}$. We show that a uniformly random degree-$n$ cover $X_n$ of $X$ has no eigenvalues below $λ_0(\widetilde{X})-\varepsilon$ other than those of $X$ and with the same multiplicity, with probability tending to $1$ as $n\to \infty$. This extends a result of Hide--Magee to metrics of pinched negative curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral gap of random covers of negatively curved noncompact surfaces Moy, Julien Spectral Theory 35P15, 58J35, 60B20 Let $(X,g)$ be a complete noncompact geometrically finite surface with pinched negative curvature $-b^2\leq K_g \leq -1$. Let $λ_0(\widetilde{X})$ denote the bottom of the $L^2-$spectrum of the Laplacian on the universal cover $\widetilde{X}$. We show that a uniformly random degree-$n$ cover $X_n$ of $X$ has no eigenvalues below $λ_0(\widetilde{X})-\varepsilon$ other than those of $X$ and with the same multiplicity, with probability tending to $1$ as $n\to \infty$. This extends a result of Hide--Magee to metrics of pinched negative curvature. |
| title | Spectral gap of random covers of negatively curved noncompact surfaces |
| topic | Spectral Theory 35P15, 58J35, 60B20 |
| url | https://arxiv.org/abs/2505.07056 |