Notes
Using the fact that any two Riemann surfaces can be mapped to each other via a quasi-conformal map that is isotopic to the identity, the distance between them is defined as the logarithm of \(K\), where \(K\) is the infimum of the maximal dilations of such quasi-conformal maps.
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Tillmann, U. Benson Farb, Richard Hain, Eduard Looijenga (Eds.): “Moduli Spaces of Riemann Surfaces”. Jahresber. Dtsch. Math. Ver. 118, 135–140 (2016). https://doi.org/10.1365/s13291-015-0127-9
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DOI: https://doi.org/10.1365/s13291-015-0127-9