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Teichmüller curves, triangle groups, and Lyapunov exponents. (English) Zbl 1203.37049
The authors construct Teichmüller curves which are uniformised by some Fuchsian triangle group commensurable to \(\Delta(m,n,\infty)\), for each \(m,n \leq \infty\). Their construction includes the Teichmüller curves obtained by Veech and C. C. Ward as special cases. Their approach uses hypergeometric differential operators. For small values of \(m\), they find billard tables which generate these Teichmüller curves. Finally, they determine the Lyapunov exponents for these Teichmüller curves.

MSC:
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
14D07 Variation of Hodge structures (algebro-geometric aspects)
30F10 Compact Riemann surfaces and uniformization
30F60 Teichmüller theory for Riemann surfaces
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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