# Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics

## TL;DR

This paper establishes asymptotic formulas for counting square-tiled surfaces with fixed foliations, linking their enumeration to Mirzakhani's results on hyperbolic geodesics, using geometric methods inspired by Mirzakhani.

## Contribution

It provides a new geometric approach to count square-tiled surfaces with prescribed foliations, connecting these counts to Mirzakhani's asymptotics for hyperbolic geodesics.

## Key findings

- Asymptotic count of square-tiled surfaces with fixed foliations
- Connection between square-tiled surface enumeration and Mirzakhani's geodesic counts
- Different approach from recent related works

## Abstract

We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.