Counting Cylinders on Z-covers of Genus 2 Square-tiled Surfaces

We count maximal cylinders on zero holonomy $\mathbb{Z}$-covers of genus $2$ square-tiled surfaces, up to $\mathbb{Z}$-action, obtaining quadratic asymptotics. We also show that the leading term of the asymptotic, called the Siegel-Veech constant, can be recovered via a large-genus approximation by intermediate finite covers. Our work applies to the infinite staircases introduced by P. Hubert and G. Weitze-Schmithüsen. For many members of this family, we explicitly compute the associated Siegel-Veech constants. In particular, we exhibit the first infinite family of examples of zero holonomy $\mathbb{Z}$-cover in which the number of cylinders grows sub-quadratically.