Mathematics – Dynamical Systems
Scientific paper
2012-03-13
Mathematics
Dynamical Systems
38 pages; comments welcome
Scientific paper
We describe the Teichm\"uller curves T(n,m) recently constructed by Bouw and M\"oller as fiberwise quotients of families of exceptionally symmetric parallelogram-tiled surfaces S(n,m) by a lift of the pillowcase symmetry group Z_2 x Z_2. Consequently, all but finitely many points (Riemann surfaces) on T(n,m) admit a parallelogram-tiled flat structure. Furthermore, the real multiplication present on a factor of Jacobians of points of T(n,m) comes from deck transformations on the S(n,m), and frequently every point on T(n,m) covers a point on some other T(n',m'). We prove that T(n,m) is always generated by Hooper's lattice surface with semiregular polygon decomposition. We compute Lyapunov exponents and determine algebraic primitivity in all cases. We give a simplified proof that the T(n,m) are Teichm\"uller curves, using a description of the period mapping in terms of Schwarz triangle mappings.
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