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

Abstract

We show that the number of square-tiled surfaces of genus \begin{document}$ g $\end{document} , with \begin{document}$ n $\end{document} marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most \begin{document}$ L $\end{document} squares, is asymptotic to \begin{document}$ L^{6g-6+2n} $\end{document} 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.