# Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes

## TL;DR

This paper explicitly calculates the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes across different strata, providing insights into their asymptotic behavior and empirical volume estimates, advancing understanding in flat surface geometry.

## Contribution

It introduces explicit formulas for one-cylinder contributions to Masur-Veech volumes and analyzes their asymptotic behavior in large genus and low-dimensional cases.

## Key findings

- The contribution is asymptotically of order 1/d in large genus.
- Explicit formulas simplify counting in large genus asymptotics.
- Empirical estimates of volumes for small-dimensional strata are provided.

## Abstract

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.