SCHWARZ TRIANGLE MAPPINGS AND TEICHMULLER CURVES: ABELIAN SQUARE-TILED SURFACES

摘要

We consider normal covers of CP~1 with abelian deck group and branched over at most four points. Families of such covers yield arithmetic Teichmuller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmuller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov exponents in terms of the period mapping. In the case of abelian square-tiled surfaces, the Lyapunov exponents are ratios of areas of hyperbolic triangles.