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On equivalence of simple closed curves in flat surfaces. (English) Zbl 1317.30062

Summary: By explicit constructions, we give direct proofs of the following results: for any distinct homotopy classes of simple closed curves \(\alpha\) and \(\beta\) in a closed surface of genus \(g > 1\), there exist a hyperbolic structure \(X\) and a holomorphic quadratic differential \(q\) on \(X\) such that \(l_{X}(\alpha) \neq l_X (\beta)\), \(\mathrm{ext}_{X}(\alpha) \neq \mathrm{ext}_{X}(\beta)\) and \(l_{q}(\alpha) \neq l_{q}(\beta)\), where \(l_{X}({\cdot})\), \(\mathrm{ext}_{X}({\cdot})\) and \(l_{q}({\cdot})\) are the hyperbolic length, the extremal length and the quadratic differential length respectively. These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.

MSC:

30F10 Compact Riemann surfaces and uniformization
57M50 General geometric structures on low-dimensional manifolds
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