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Spectral gap of random hyperbolic surfaces

Nalini Anantharaman, Laura Monk

Abstract

Let $X$ be a closed, connected, oriented surface of genus $g$, with a hyperbolic metric chosen at random according to the Weil--Petersson measure on the moduli space of Riemannian metrics. Let $λ_1=λ_1(X)$ bethe first non-zero eigenvalue of the Laplacian on $X$ or, in other words, the spectral gap.In this paper we give a full road-map to prove that for arbitrarily small~$α>0$,\begin{align*} \Pwp{λ_1 \leq \frac{1}{4} - α^2 } \Lim_{g\To +\infty} 0.\end{align*}The full proofs are deferred to separate papers.

Spectral gap of random hyperbolic surfaces

Abstract

Let be a closed, connected, oriented surface of genus , with a hyperbolic metric chosen at random according to the Weil--Petersson measure on the moduli space of Riemannian metrics. Let bethe first non-zero eigenvalue of the Laplacian on or, in other words, the spectral gap.In this paper we give a full road-map to prove that for arbitrarily small~,\begin{align*} \Pwp{λ_1 \leq \frac{1}{4} - α^2 } \Lim_{g\To +\infty} 0.\end{align*}The full proofs are deferred to separate papers.
Paper Structure (37 sections, 25 theorems, 97 equations, 2 figures)

This paper contains 37 sections, 25 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Walkthrough of the proof
  3. The trace method
  4. The average contribution of simple geodesics
  5. Killing dominant exponential terms
  6. Volume functions for local types, and the Friedman--Ramanujan property
  7. Local topological types of loops
  8. Volume functions
  9. Asymptotic expansions in inverse powers of $g$
  10. The Friedman--Ramanujan property and integration by parts argument
  11. Tangles and how to remove them
  12. The failure of the trace method
  13. Tangles and the exponential proliferation of local types
  14. Inclusion-exclusion and Moebius formula
  15. How to prove the Friedman--Ramanujan property
  16. ...and 22 more sections

Key Result

Lemma 2.1

Let $\alpha \in (0, \frac{1}{2})$. For any $0 < \epsilon < \frac{1}{4} - \alpha^2$, there exists a constant $C_{\alpha,\epsilon} > 0$ such that, for any hyperbolic surface $X$, any $L \geq 1$,

Figures (2)

  • Figure 1: Examples of local topological types. From left to right, we have the type "simple", a figure-eight, and then two generalized eights (see § \ref{['s:gene_eight']}).
  • Figure 2: A generalized eight with $r=3$ intersections, and the segments $B_k^+$ and $\mathcal{I}_j^\pm$ in its diagram.

Theorems & Definitions (48)

  • Lemma 2.1: Ours1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.5: Ours1
  • Lemma 2.6: Ours1, adapted from buser1992
  • Lemma 2.7: Ours1
  • Remark 2.8
  • Definition 2.10
  • Definition 2.11
  • Theorem 2.13: Ours1, Theorem 5.5
  • ...and 38 more