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Spectral gap of random covers of negatively curved noncompact surfaces

Julien Moy

TL;DR

This work proves that for a complete noncompact geometrically finite surface with pinched negative curvature, a uniformly random degree-$n$ cover carries no new $L^2$ eigenvalues below $\lambda_0(\tilde{X})-\varepsilon$ with high probability as $n$ grows. The authors decompose the heat operator into end and interior parts, using strong convergence of random permutation representations to compare the interior dynamics with the universal cover and identify the spectral contribution with the base spectrum. The end dynamics are controlled by the essential spectrum via end-wise Dirichlet problems, enabling a uniform bound tied to $\lambda_0(\tilde{X})$. The result extends Hide–Magee’s gap phenomenon to variable negative curvature and cements the role of strong representation theory in geometric spectral problems for random covers.

Abstract

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.

Spectral gap of random covers of negatively curved noncompact surfaces

TL;DR

This work proves that for a complete noncompact geometrically finite surface with pinched negative curvature, a uniformly random degree- cover carries no new eigenvalues below with high probability as grows. The authors decompose the heat operator into end and interior parts, using strong convergence of random permutation representations to compare the interior dynamics with the universal cover and identify the spectral contribution with the base spectrum. The end dynamics are controlled by the essential spectrum via end-wise Dirichlet problems, enabling a uniform bound tied to . The result extends Hide–Magee’s gap phenomenon to variable negative curvature and cements the role of strong representation theory in geometric spectral problems for random covers.

Abstract

Let be a complete noncompact geometrically finite surface with pinched negative curvature . Let denote the bottom of the spectrum of the Laplacian on the universal cover . We show that a uniformly random degree- cover of has no eigenvalues below other than those of and with the same multiplicity, with probability tending to as . This extends a result of Hide--Magee to metrics of pinched negative curvature.
Paper Structure (21 sections, 15 theorems, 135 equations)

This paper contains 21 sections, 15 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Overview of the proof
  4. Geometric preliminaries
  5. Comparison estimates
  6. Smooth cutoff functions
  7. Dirichlet fundamental domains
  8. Laplacian and heat kernel on Riemannian manifolds
  9. The Dirichlet Laplacian
  10. Bottom of the spectrum on finite covers
  11. The heat kernel
  12. Heat kernel upper bounds
  13. Heat operator in the ends
  14. Proof of Theorem \ref{['thm: main']}
  15. Strong convergence
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any $\varepsilon>0$, with probability tending to $1$ as $n\to +\infty$, one has where the multiplicities coincide on both sides.

Theorems & Definitions (23)

  • Theorem 1.1
  • Lemma 2.1: jost1984harmonic
  • Lemma 2.2: Decay of injectivity radius
  • Remark 2.3
  • Lemma 2.4: Volume comparison
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.7: HM
  • ...and 13 more