Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral Gaps on Large Hyperbolic Surfaces

Laura Monk, Frédéric Naud

TL;DR

This survey traces the evolution of understanding the spectral gap $\lambda_1$ for large-volume hyperbolic surfaces, linking geometric, probabilistic, and analytic methods. It covers historical bounds via Cheeger-type arguments, trace formulas, and three principal random models (Brooks–Makover, Weil–Petersson, and random covers), highlighting barriers to achieving the conjectured $\tfrac{1}{4}$-gap. A central theme is the rise of strong convergence and the polynomial method, which, together with probabilistic and geometric tools, establishes near-optimal or optimal gap results in broad random settings and across finite- and infinite-volume regimes. The developments reveal deep connections between hyperbolic geometry, random graph theory, and representation theory, enabling robust control of spectral data in large-genus surfaces and their covers with wide potential applications.

Abstract

In this expository paper, we review the history and the recent breakthroughs in the spectral theory of large volume hyperbolic surfaces. More precisely, we focus mostly on the investigation of the first non-trivial eigenvalue $λ_1$ and its possible behaviour in the large volume regime.

Spectral Gaps on Large Hyperbolic Surfaces

TL;DR

This survey traces the evolution of understanding the spectral gap for large-volume hyperbolic surfaces, linking geometric, probabilistic, and analytic methods. It covers historical bounds via Cheeger-type arguments, trace formulas, and three principal random models (Brooks–Makover, Weil–Petersson, and random covers), highlighting barriers to achieving the conjectured -gap. A central theme is the rise of strong convergence and the polynomial method, which, together with probabilistic and geometric tools, establishes near-optimal or optimal gap results in broad random settings and across finite- and infinite-volume regimes. The developments reveal deep connections between hyperbolic geometry, random graph theory, and representation theory, enabling robust control of spectral data in large-genus surfaces and their covers with wide potential applications.

Abstract

In this expository paper, we review the history and the recent breakthroughs in the spectral theory of large volume hyperbolic surfaces. More precisely, we focus mostly on the investigation of the first non-trivial eigenvalue and its possible behaviour in the large volume regime.
Paper Structure (26 sections, 10 theorems, 45 equations, 3 figures)

This paper contains 26 sections, 10 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction: from Selberg to Mirzakhani, the spectral gap of hyperbolic surfaces.
  2. Spectral gap: history, expected behaviour, interactions.
  3. Definition.
  4. Link with geodesic counting.
  5. Link with connectivity.
  6. Large-volume limit.
  7. Benjamini--Schramm convergence.
  8. Random regular graphs are optimal expanders.
  9. Models of random hyperbolic surfaces and their spectral gap.
  10. First families of expanders: the Cheeger method.
  11. The Brooks--Makover model.
  12. The Weil--Petersson model.
  13. The sample space.
  14. Parametrizations.
  15. The Weil--Petersson symplectic form.
  16. ...and 11 more sections

Key Result

Theorem 3

There exists $C>0$ such that $\lambda_1(X) \geq C$ with high probability for the Brooks--Makover model.

Figures (3)

  • Figure 1: Fenchel--Nielsen coordinates in genus $g=3$.
  • Figure 2: Limit set of Schottky group $\Gamma$ and pair of pants with funnels as quotient surface $X$.
  • Figure 3: Two long closed geodesics on a large-genus surface.

Theorems & Definitions (13)

  • Conjecture 1
  • Conjecture 2
  • Theorem 3: brooks2004
  • Theorem 4: Wolpert, wolpert1981
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Definition 9
  • Theorem 10
  • ...and 3 more