{"title":"A random cover of a compact hyperbolic surface has relative spectral gap $$\\frac{3}{16}-\\varepsilon $$ 3 16 - ε","authors":"Michael Magee, Frédéric Naud, Doron Puder","doi":"10.1007/s00039-022-00602-x","DOIUrl":null,"url":null,"abstract":"<p>Let <i>X</i> be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature <span>\\(-1\\)</span>. For each <span>\\(n\\in {\\mathbf {N}}\\)</span>, let <span>\\(X_{n}\\)</span> be a random degree-<i>n</i> cover of <i>X</i> sampled uniformly from all degree-<i>n</i> Riemannian covering spaces of <i>X</i>. An eigenvalue of <i>X</i> or <span>\\(X_{n}\\)</span> is an eigenvalue of the associated Laplacian operator <span>\\(\\Delta _{X}\\)</span> or <span>\\(\\Delta _{X_{n}}\\)</span>. We say that an eigenvalue of <span>\\(X_{n}\\)</span> is <i>new </i>if it occurs with greater multiplicity than in <i>X</i>. We prove that for any <span>\\(\\varepsilon >0\\)</span>, with probability tending to 1 as <span>\\(n\\rightarrow \\infty \\)</span>, there are no new eigenvalues of <span>\\(X_{n}\\)</span> below <span>\\(\\frac{3}{16}-\\varepsilon \\)</span>. We conjecture that the same result holds with <span>\\(\\frac{3}{16}\\)</span> replaced by <span>\\(\\frac{1}{4}\\)</span>.\n</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.4000,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"10","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometric and Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00039-022-00602-x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 10
Abstract
Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature \(-1\). For each \(n\in {\mathbf {N}}\), let \(X_{n}\) be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or \(X_{n}\) is an eigenvalue of the associated Laplacian operator \(\Delta _{X}\) or \(\Delta _{X_{n}}\). We say that an eigenvalue of \(X_{n}\) is new if it occurs with greater multiplicity than in X. We prove that for any \(\varepsilon >0\), with probability tending to 1 as \(n\rightarrow \infty \), there are no new eigenvalues of \(X_{n}\) below \(\frac{3}{16}-\varepsilon \). We conjecture that the same result holds with \(\frac{3}{16}\) replaced by \(\frac{1}{4}\).
期刊介绍:
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