Schwarz triangle mappings and Teichmüller curves I: abelian square-tiled surfaces

Details Schwarz triangle mappings and Teichmüller curves I: abelian square-tiled surfaces Schwarz triangle mappings and Teichmüller curves I: abelian square-tiled surfaces

2012-03-13

arxiv.org/abs/1203.2683v1

Mathematics

Dynamical Systems

24 pages; comments welcome

Scientific paper

Abelian parallelogram-tiled surfaces are normal covers of CP^1 with abelian deck group, branched over at most four points and equipped with a lift of a flat metric on CP^1 tiled by two isometric parallelograms. Families of such surfaces yield arithmetic Teichm\"uller curves, whose period mapping may be described in terms of Schwarz triangle mappings. We prove a formula which, in appropriate situations, computes Lyapunov exponents from the period mapping. Applied to abelian square-tiled surfaces, this formula gives all the individual Lyapunov exponents and clarifies their geometric significance. We also give a combinatorial model for abelian square-tiled surfaces. Together with the description of the period map, this will be used in the sequel to construct and study the non-arithmetic Veech-Ward-Bouw-M\"oller Teichm\"uller curves.

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