Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials

Abstract

We present an approach for counting the Teichmuller volumes of the moduli spaces of Abelian differentials on a Riemann surface of genus g. We show that the volumes can be counted by means of counting the “integer points” in the corresponding moduli space. The “integer points” are represented by square tiled surfaces — the flat surfaces tiled by unit squares. Such tilings have several conical singularities with 8, 12,... adjacent unit squares. Counting the leading term in the asymptotics of the number of tilings having at most N unit squares, we get the volumes of the corresponding strata of the moduli spaces.