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

TL;DR

This paper provides an effective counting estimate with a power-saving error term for square-tiled surfaces with specified foliations within connected strata, improving previous asymptotic formulas.

Contribution

It introduces a precise counting method with an error estimate for square-tiled surfaces in specified foliation classes, enhancing prior asymptotic results.

Findings

Effective estimate with power-saving error term for square-tiled surfaces

Counts surfaces with prescribed foliations in connected strata

Strengthens previous asymptotic counting formulas

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.