Typical hyperbolic surfaces have an optimal spectral gap
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912838312787968 |
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| author | Monk, Laura |
| author_facet | Monk, Laura |
| contents | The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus $g$, equivalently), we know that the spectral gap is asymptotically bounded above by $\frac 14$. The aim of these talks is to present joint work with Nalini Anantharaman, where we prove that most hyperbolic surfaces have a near-optimal spectral gap. That is to say, we prove that, for any $ε> 0$, the Weil--Petersson probability for a hyperbolic surface of genus $g$ to have a spectral gap greater than $\frac 14- ε$ goes to one as $g$ goes to infinity. This statement is analogous to Alon's 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which shares many similarities with Friedman's work, and introduce new tools and ideas that we have developed in order to tackle this problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_15157 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Typical hyperbolic surfaces have an optimal spectral gap Monk, Laura Spectral Theory Metric Geometry Primary 58J50, 32G15. Secondary 05C80, 11F72 The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus $g$, equivalently), we know that the spectral gap is asymptotically bounded above by $\frac 14$. The aim of these talks is to present joint work with Nalini Anantharaman, where we prove that most hyperbolic surfaces have a near-optimal spectral gap. That is to say, we prove that, for any $ε> 0$, the Weil--Petersson probability for a hyperbolic surface of genus $g$ to have a spectral gap greater than $\frac 14- ε$ goes to one as $g$ goes to infinity. This statement is analogous to Alon's 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which shares many similarities with Friedman's work, and introduce new tools and ideas that we have developed in order to tackle this problem. |
| title | Typical hyperbolic surfaces have an optimal spectral gap |
| topic | Spectral Theory Metric Geometry Primary 58J50, 32G15. Secondary 05C80, 11F72 |
| url | https://arxiv.org/abs/2601.15157 |