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Random covers of surfaces

Sophie Wright

TL;DR

The paper studies random covers of a closed hyperbolic surface by focusing on conjugacy classes of free subgroups Δ ≅ F_k of π1(Σ) and introducing a length via minimal carrier graphs. It proves a Huber-type asymptotic for the number of such subgroups with length ≤ L and constructs counting measures that converge, in the weak-* sense, to a surface‑independent Lebesgue-class measure on the moduli space of volume-one metric graphs 𝓜_k; a Patterson–Sullivan variant yields the same limit. The limit measure is explicitly described as a weighted sum over graph types, and this framework enables the computation of asymptotics for geometric invariants of random covers, with explicit results for k=2 such as the expected systole ~ (23/90)L and statistics for separating orthogeodesics and topological types. The work connects random subgroups to the geometry of metric graph moduli space, providing a probabilistic understanding of typical covers independent of the underlying surface.

Abstract

We study random covers of a closed hyperbolic surface $Σ$, subject to the condition that, for $k\geq 2$, the fundamental group is isomorphic to the free group $F_k$. We show that asymptotically they distribute according to a specific probability measure on the moduli space of metric graphs. As we will demonstrate with explicit calculations for $k=2$, this allows us to determine asymptotic values for the expectation of the systole and other geometric invariants of the covers.

Random covers of surfaces

TL;DR

The paper studies random covers of a closed hyperbolic surface by focusing on conjugacy classes of free subgroups Δ ≅ F_k of π1(Σ) and introducing a length via minimal carrier graphs. It proves a Huber-type asymptotic for the number of such subgroups with length ≤ L and constructs counting measures that converge, in the weak-* sense, to a surface‑independent Lebesgue-class measure on the moduli space of volume-one metric graphs 𝓜_k; a Patterson–Sullivan variant yields the same limit. The limit measure is explicitly described as a weighted sum over graph types, and this framework enables the computation of asymptotics for geometric invariants of random covers, with explicit results for k=2 such as the expected systole ~ (23/90)L and statistics for separating orthogeodesics and topological types. The work connects random subgroups to the geometry of metric graph moduli space, providing a probabilistic understanding of typical covers independent of the underlying surface.

Abstract

We study random covers of a closed hyperbolic surface , subject to the condition that, for , the fundamental group is isomorphic to the free group . We show that asymptotically they distribute according to a specific probability measure on the moduli space of metric graphs. As we will demonstrate with explicit calculations for , this allows us to determine asymptotic values for the expectation of the systole and other geometric invariants of the covers.

Paper Structure

This paper contains 6 sections, 18 theorems, 104 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Spaces of conjugacy classes of subgroups
  3. Counting conjugacy classes of subgroups
  4. Convergence of probability measures
  5. Patterson-Sullivan analogue
  6. Expected properties of covers

Key Result

Theorem 1.1

For a closed, orientable hyperbolic surface $\Sigma$ of genus $g$ and for any $k\geq 2$, the cardinality of the set $\mathbf{G}_{\Sigma,k}(L)$ has asymptotic growth: as $L\to\infty$, where $\operatorname{Typ_k}$ is the set of homeomorphism classes of connected trivalent graphs of rank $k$ and $\operatorname{Aut}(X)$ is the set of automorphisms of the topological graph $X$.

Figures (7)

  • Figure 1: The image of a rank $2$ carrier graph on a genus 3 surface. The underlying graph is the dumbbell
  • Figure 2: Moving a trivalent vertex so that all angles are equal to $\frac{2\pi}{3}$ reduces the overall length
  • Figure 3: Splitting a vertex with degree $>3$ reduces the overall length
  • Figure 4: Tiling of the hyperbolic plane by heptagons with angles $\frac{2\pi}{3}$, made with 'hypertiling' hyptiling
  • Figure 5: The two rank-2 trivalent graphs: the dumbbell graph and the theta graph.
  • ...and 2 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 28 more