5 Siegel – Veech Constants in H ( 2 )

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 generically has quadratic asymptotics in this length. Siegel–Veech constants are coefficients of these quadratic growth rates, and coincide for almost all surfaces in each moduli space of translation surfaces. Square-tiled surfaces are some specific translation surfaces whose Siegel–Veech do not equal the generic ones. It is an interesting question whether, as n tends to infinity, the Siegel–Veech constants of square-tiled surfaces with n tiles tend to the generic constants of the ambient moduli space. Here we prove that it is the case in the moduli space H(2) of translation surfaces of genus two with one singularity.