A random cover of a compact hyperbolic surface has relative spectral gap 316-ε

• Published in: Geometric and functional analysis Vol. 32; no. 3; pp. 595 - 661
• Main Authors: Magee, Michael, Naud, Frédéric, Puder, Doron
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2022; Springer Nature B.V
• Subjects: Analysis; Eigenvalues; Mathematics; Mathematics and Statistics
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-022-00602-x

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 ∈ 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 Δ X or Δ 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 ε > 0 , with probability tending to 1 as n → ∞ , there are no new eigenvalues of X n below 3 16 - ε . We conjecture that the same result holds with 3 16 replaced by 1 4 .