The Weil-Petersson distance between finite degree covers of punctured Riemann surfaces and random ideal triangulations

Jeudi 22 mai 2008 14:00-15:00 - Markovic Vladimir - Univ. of Warwick

Résumé : A Riemann surface is said to be of finite type (g,n) if it arises from a closed Riemann surface of genus g by removing n points.
It is well known that every two finite type Riemann surfaces have a common branched holomorphic cover, but usually there is no common unbranched cover. Given two finite type surfaces S and R (that are either both closed or both open) the question is whether we can find finite degree unbranched covers S’-> S and R’ -> R, such that the Riemann surfaces S’ and R’ are homeomorphic and « close » to each other in the corresponding Moduli space, with respect to some natural metric on this Moduli space (like the Teichmuller or the Weil-Petersson metric). We show that such covers S’ and R’ exist if both S and R are open finite type Riemann surfaces. We explicitly construct the covers S’ and R’ so that the Weil-Petersson distance between S’ and R’ is arbitrary small. Our method is to construct certain ideal triangulations of the hyperbolic plane that are associated to measures on the unit tangent bundle of a given Riemann surface.
This is a joint work with Jeremy Kahn.

Lieu : bât. 425 - 121-123

The Weil-Petersson distance between finite degree covers of punctured Riemann surfaces and random ideal triangulations  Version PDF