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

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