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On the spectral gap of negatively curved surface covers

Will Hide, Julien Moy, Frederic Naud

TL;DR

This work analyzes the Laplacian spectrum of random n-sheeted covers of a negatively curved compact surface X. It develops a robust heat kernel framework on covers to replace Selberg trace formulas in variable curvature, combining Gaussian heat kernel bounds with Green kernel techniques and Ancona–Gouëzel inequalities. The authors prove that with high probability, no new eigenvalues occur below a threshold λ0/2 minus ε, delivering an explicit spectral gap bound tied to the universal cover bottom λ0 and the geodesic flow entropy δ, and formulate a conjecture for the optimal gap. They further show near optimal gaps via strong convergence of representations, using Louder–Magee results and strong convergence to relate covers to the regular representation. The approach blends geometric group theory, heat kernel analysis, and probabilistic representations to extend spectral gap results from constant to variable negative curvature, offering new counting bounds and a robust analytic toolkit for random covers.

Abstract

Given a negatively curved compact Riemannian surface $X$, we give an explicit estimate, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of $X$. The explicit gap is given in terms of the bottom of the spectrum of the universal cover of $X$ and the topological entropy of the geodesic flow on X. This result generalizes in variable curvature a result of Magee-Naud-Puder for hyperbolic surfaces. We then formulate a conjecture on the optimal spectral gap and show that there exists covers with near optimal spectral gaps using a result of Louder-Magee and techniques of strong convergence from random matrix theory.

On the spectral gap of negatively curved surface covers

TL;DR

This work analyzes the Laplacian spectrum of random n-sheeted covers of a negatively curved compact surface X. It develops a robust heat kernel framework on covers to replace Selberg trace formulas in variable curvature, combining Gaussian heat kernel bounds with Green kernel techniques and Ancona–Gouëzel inequalities. The authors prove that with high probability, no new eigenvalues occur below a threshold λ0/2 minus ε, delivering an explicit spectral gap bound tied to the universal cover bottom λ0 and the geodesic flow entropy δ, and formulate a conjecture for the optimal gap. They further show near optimal gaps via strong convergence of representations, using Louder–Magee results and strong convergence to relate covers to the regular representation. The approach blends geometric group theory, heat kernel analysis, and probabilistic representations to extend spectral gap results from constant to variable negative curvature, offering new counting bounds and a robust analytic toolkit for random covers.

Abstract

Given a negatively curved compact Riemannian surface , we give an explicit estimate, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of . The explicit gap is given in terms of the bottom of the spectrum of the universal cover of and the topological entropy of the geodesic flow on X. This result generalizes in variable curvature a result of Magee-Naud-Puder for hyperbolic surfaces. We then formulate a conjecture on the optimal spectral gap and show that there exists covers with near optimal spectral gaps using a result of Louder-Magee and techniques of strong convergence from random matrix theory.

Paper Structure

This paper contains 21 sections, 19 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Heat Kernels on random covers
  3. Geometric preliminaries
  4. Reminders on the isometry group $\Gamma$
  5. Closed geodesics and conjugacy classes of $\Gamma$
  6. Primitive elements of $\Gamma$
  7. Counting results
  8. Comparison results
  9. Proof of Theorem \ref{['main1']}
  10. Strategy of the proof
  11. The main combinatorial estimate
  12. The contribution of primitive elements
  13. Estimating $(\mathrm{I})$
  14. Estimating $(\mathrm{II})$
  15. The contribution of non-primitive elements
  16. ...and 6 more sections

Key Result

Theorem 1.1

For all $\varepsilon>0$, with high probability as $n\rightarrow +\infty$, we have where the multiplicities coincide on both sides.

Figures (1)

  • Figure 1: The shaded area represents the fundamental domain $\mathcal{F}_\gamma$ for the action of $\langle \gamma\rangle$ on $\widetilde{X}$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • ...and 30 more