## 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∈ 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 ΔXn. 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 316-ε. We conjecture that the same result holds with 316 replaced by 14.

Original language | English |
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Pages (from-to) | 595-661 |

Number of pages | 67 |

Journal | Geometric and Functional Analysis |

Volume | 32 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2022 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology