  We consider normal covers of CP^1 with abelian deck group, branched over at
most four points. Families of such covers yield arithmetic Teichm\"uller
curves, whose period mapping may be described geometrically in terms of Schwarz
triangle mappings. These Teichm\"uller 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.
