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Counting geodesics on expander surfaces

Benjamin Dozier, Jenya Sapir

TL;DR

The paper investigates the structure of long closed geodesics on expander hyperbolic surfaces of large genus, proving that almost all geodesics with length above $c\sqrt{g}\log g$ are non-simple and that almost all with length above $c g(\log g)^2$ are filling. By transferring Margulis' counting approach to low length scales via flow boxes and employing a uniform spectral gap to obtain effective mixing and multiple mixing, the authors derive an effective prime geodesic theorem and robust global counts from local box-counts. This framework yields sharp bounds on the prevalence of simple geodesics and establishes filling as the typical behavior for sufficiently long geodesics. The results apply to Weil-Petersson random surfaces, random covers, and Brooks-Makover random surfaces, and the techniques bridge graph-based counting with geometric flow dynamics, providing tools for analyzing random high-genus hyperbolic surfaces with uniform spectral gaps.

Abstract

We study properties of typical closed geodesics on expander surfaces of high genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the Laplacian. Under an additional systole lower bound assumption, we show almost every geodesic of length much greater than $\sqrt{g}\log g$ is non-simple. And we prove almost every closed geodesic of length much greater than $g (\log g)^2$ is filling, i.e. each component of the complement of the geodesic is a topological disc. Our results apply to Weil-Petersson random surfaces, random covers of a fixed surface, and Brooks-Makover random surfaces, since these models are known to have uniform spectral gap asymptotically almost surely. Our proof technique involves adapting Margulis' counting strategy to work at low length scales.

Counting geodesics on expander surfaces

TL;DR

The paper investigates the structure of long closed geodesics on expander hyperbolic surfaces of large genus, proving that almost all geodesics with length above are non-simple and that almost all with length above are filling. By transferring Margulis' counting approach to low length scales via flow boxes and employing a uniform spectral gap to obtain effective mixing and multiple mixing, the authors derive an effective prime geodesic theorem and robust global counts from local box-counts. This framework yields sharp bounds on the prevalence of simple geodesics and establishes filling as the typical behavior for sufficiently long geodesics. The results apply to Weil-Petersson random surfaces, random covers, and Brooks-Makover random surfaces, and the techniques bridge graph-based counting with geometric flow dynamics, providing tools for analyzing random high-genus hyperbolic surfaces with uniform spectral gaps.

Abstract

We study properties of typical closed geodesics on expander surfaces of high genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the Laplacian. Under an additional systole lower bound assumption, we show almost every geodesic of length much greater than is non-simple. And we prove almost every closed geodesic of length much greater than is filling, i.e. each component of the complement of the geodesic is a topological disc. Our results apply to Weil-Petersson random surfaces, random covers of a fixed surface, and Brooks-Makover random surfaces, since these models are known to have uniform spectral gap asymptotically almost surely. Our proof technique involves adapting Margulis' counting strategy to work at low length scales.
Paper Structure (42 sections, 39 theorems, 205 equations, 9 figures)

This paper contains 42 sections, 39 theorems, 205 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Asymptotic notation.
  3. Applications to random surfaces
  4. Weil-Petersson random surfaces
  5. Random covers
  6. Brooks-Makover (Belyi) random surfaces
  7. Relation to prior work
  8. Discussion and outline of proof
  9. Outline of proof.
  10. Acknowledgements
  11. General results on flow boxes and counting
  12. Notation and setup
  13. Matrices for geodesic and horocylic flows.
  14. Flow boxes $B(v)$.
  15. Foliation by geodesics.
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $\delta, s_0, \epsilon>0$. There exists a constant $c=c(\delta,s_0,\epsilon)$ such that for any $\delta$-expander surface $X$ of genus $g$ with systole at least $s_0$, and any $L> c\sqrt{g} \log g$,

Figures (9)

  • Figure 1: The geometric mechanism corresponding to the "Markov property" $\mu'(P\cap F)=\mu'(P)\mu'(F)$ in \ref{['lem:band-mix-aux']}. The components of $F$ are full width in the contracting direction, corresponding to a condition on future behavior. The subbox $P$ is full width in the contracting direction, which could come from conditioning on past behavior. The future and past are independent. (The geodesic flow direction has been suppressed in the diagram.)
  • Figure 2: \ref{['lem:band-future']} and its proof.
  • Figure 3: The geometric mechanism that allows one to prove (effective) multiple mixing, using mixing. In the middle box, the red components, coming from a condition on the past, are "perpendicular" to the blue components, which come from a condition on the future (compare this middle box to \ref{['fig:band_mix']}). This is the analog in the hyperbolic dynamics setting of the Markov property for random walks.
  • Figure 4: Transverse flow boxes guaranteeing a self-intersection. In (the unit tangent bundle over) every pair of pants, we can fit a pair of transverse flow boxes $B,\hat{B}$ of definite size. Any geodesic that lands in both $B$ and $\hat{B}$ must have a self-intersection. Thus a simple geodesic cannot hit both boxes in any such pair. We use this condition to bound the number of simple closed geodesics of length at most $L$.
  • Figure 5: Every right-angled hyperbolic hexagon contains a disc of definite area.
  • ...and 4 more figures

Theorems & Definitions (88)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6: Weil-Petersson surfaces
  • proof
  • Corollary 1.7: Random covers
  • proof
  • Corollary 1.8: Brooks-Makover surfaces
  • ...and 78 more