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Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata

Francisco Arana-Herrera

TL;DR

The paper proves an effective, power-saving count for square-tiled surfaces in a connected component of a stratum of quadratic differentials with prescribed vertical and horizontal foliations. It builds on connections between square-tiled counts and simple closed curves, and leverages the Eskin–Mirzakhani–Mohammadi (EMM19) framework to obtain an explicit leading term and error for all components, not just the principal one. The approach relies on moderately slanted cylinder diagrams and cylinder train tracks to parametrize horospherical sets and train-track coordinates, together with the exponential mixing of the Teichmüller flow. This yields a robust method for effective counting in moduli spaces of quadratic differentials with broader applicability than prior ergodic-based approaches.

Abstract

We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most $L$ squares. This result strengthens asymptotic counting formulas in work of Delecroix, Goujard, Zograf, Zorich, and the author.

Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata

TL;DR

The paper proves an effective, power-saving count for square-tiled surfaces in a connected component of a stratum of quadratic differentials with prescribed vertical and horizontal foliations. It builds on connections between square-tiled counts and simple closed curves, and leverages the Eskin–Mirzakhani–Mohammadi (EMM19) framework to obtain an explicit leading term and error for all components, not just the principal one. The approach relies on moderately slanted cylinder diagrams and cylinder train tracks to parametrize horospherical sets and train-track coordinates, together with the exponential mixing of the Teichmüller flow. This yields a robust method for effective counting in moduli spaces of quadratic differentials with broader applicability than prior ergodic-based approaches.

Abstract

We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most squares. This result strengthens asymptotic counting formulas in work of Delecroix, Goujard, Zograf, Zorich, and the author.
Paper Structure (3 sections, 18 theorems, 43 equations, 5 figures)

This paper contains 3 sections, 18 theorems, 43 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Horospheres in connected components of strata
  3. Counting square-tiled surfaces

Key Result

Theorem 1.1

Let $\mathcal{Q}$ be a connected component of a stratum of quadratic differentials of complex dimension $h > 0$ and $\gamma_1$, $\gamma_2$ be a pair of integral simple closed multi-curves on the corresponding topological surface. Then, there exist positive constants $v(\gamma_1,\mathcal{Q}) > 0$, $v

Figures (5)

  • Figure 1: Example of a square-tiled surface of genus $2$ with two zeroes of order $2$. The horizontal core multi-curve is $\alpha_1 + 2 \alpha_2$. The vertical core multi-curve is $\beta_1 + \beta_2 + \beta_3$.
  • Figure 2: Cylinder diagram of a quadratic differential in $\mathcal{Q}(4,0)$. The horizontal foliation is identified with a simple closed curve on a genus $0$ surface with $4$ punctures that separates the punctures into two sets of two. This is a moderately slanted cylinder diagram.
  • Figure 3: Triangulation associated to the moderately slanted cylinder diagram in Figure \ref{['fig:cyl_diag']}.
  • Figure 4: The $1$-complexes in a triangle.
  • Figure 5: Train track associated to the moderately slanted cylinder diagram in Figure \ref{['fig:cyl_diag']}.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • ...and 15 more