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 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.

Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

“…Our announcement of the explicit relation between flat and hyperbolic counts described in the current paper inspired F. Arana-Herrera to suggest in [AH1] an alternative geometric proof of these results in the spirit of M. Mirzakhani.…”

“…However, Mirzakhani does not give any formula for the value of the normalization constant presented in (1.34). This constant was recently computed by F. Arana-Herrera [AH1] and by L. Monin and I. Telpukhovkiy [MoT] simultaneously and independently of us by different methods. The same value of the constant in (1.34) is obtained by V. Erlandsson and J. Souto in [ErSo] through an approach different from all the ones mentioned above.…”

Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow

“…The integral points of the PIL structure on SH + (λ) correspond to integer multicurves transverse to λ, so when λ is itself a multicurve, integral points correspond to square-tiled surfaces. Using our coordinates for F uu (λ), one can recover the leading coefficient for the polynomial counting the number of square-tiled surfaces with given horizontal curve of bounded area (which was originally computed in [AH20a,DGZZ20]). In particular, since renormalized lattice point counts equidistribute to Lebesgue measure, the coefficient in question can be identified as the Lebesgue measure of (a torus bundle over) the portion of the combinatorial moduli space B(S \ λ)/ Mod(S \ λ) with controlled boundary lengths.…”

A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

“…Remarkably, this probability is computable and does not depend on X! Even more remarkably, recent discoveries prove that the same probabilities appear in discrete problems about surfaces assembled out of finitely many unit squares [DGZZ,AH19].…”