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

2008/06/13 by Kahn, Jeremy, Markovic, Vladimir · 1 citation
#30F60 #32G15 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0806.2304

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.

Cited by

• Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves

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