Random ideal triangulations and the Weil-Petersson distance between finite degree covers of punctured Riemann surfaces

Jeremy Kahn; Vladimir Markovic

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arXiv · 0806.2304

Random ideal triangulations and the Weil-Petersson distance between finite degree covers of punctured Riemann surfaces

Jeremy Kahn·Vladimir Markovic

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Abstract

We prove that any two finite-area non-compact hyperbolic Riemann surfaces S and T have finite covers that are arbitrarily close in the normalized Weil-Petersson metric, where we normalize by dividing the square of the metric by the area of the surface. In the case where T is the modular surface this reduces to showing that S has a finite cover with a proper ideal triangulation where most of the shear coordinates are small; we will construct such a cover out of a random collection of immersed ideal triangles in S.

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Jeremy Kahn, Vladimir Markovic. 2008-06-13. Random ideal triangulations and the Weil-Petersson distance between finite degree covers of punctured Riemann surfaces. https://arxiv.org/abs/0806.2304

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