Abstract
We briefly describe another probabilistic model, the random covering model of a hyperbolic surface, in order to be able to state the Magee-Naud-Puder and Magee-Hide theorems concerning the spectral hole of these surfaces.
Finally, we turn to the description of the Weill-Petersson measure on the moduli space of hyperbolic structures, on a compact oriented surface.
As this moduli space is defined as a quotient of the Teichmüller space, we'll need to develop some basic formulas for integration on quotient spaces.
References
M. Magee, F. Naud, D. Puder, A random cover of a compact hyperbolic surface has relative spectral gap 3/16-epsilon. Geometric and functional analysis 2022, Theorem 1.5
W. Hide, M. Magee, Near optimal spectral gaps for hyperbolic surfaces, Annals of Math. 2023, Theorem 1.1
P. Buser, Geometry and Spectra of Compact Riemann Surfaces, beginning of Chapter 6
S. Wolpert, On the Weil-Petersson geometry of the moduli space of curves, American J. of Math. 1985
