Siegel–Veech constants in $\mathcal{H}(2)$

Abstract

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length was proved by Eskin and Masur to generically have quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics called a Siegel–Veech constant which is shared by almost all surfaces in each moduli space of translation surfaces.

Square-tiled surfaces are specific translation surfaces which have their own quadratic asymptotics for the number of cylinders of closed geodesics. It is an interesting question whether the Siegel–Veech constant of a given moduli space can be recovered as a limit of individual constants of square-tiled surfaces in this moduli space. We prove that this is the case in the moduli space $\mathcal{\mathscr{H}}\left(2\right)$ of translation surfaces of genus two with one singularity.