I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete examples of translation surfaces. For example, if I take four copies of equilateral triangle and glue them together side by side to form a tetrahedron, how do I put a Riemann surface structure on it ? I want some detailed explanations etc. Could you cite any reference(s) ?
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People normally take the definition: translation surface = pair (X,w), where X is Riemann Surface and w is a non identically zero holomorphic form on X. Strictly, if you want to make a translation surface out of this pair you have to remove the zeros of w from X and then integrate w. As a general reference take the book: Flat surfaces (by Anton Zorich) in collection "Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems'', P. Cartier; B. Julia; P. Moussa; P. Vanhove (Editors), Springer-Verlag, Berlin, 2006, 439-586. |
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There is a huge literature, and I'm not sure exactly what you are looking for. That being said, Masur-Tabachnikov's survey "Rational billiards and flat structures" and Masur's survey "Ergodic Theory of Translation surfaces" contain a lot of introductory material. Both can be found on Masur's webpage here. |
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