Geometry Seminar

For a compact Riemann surface S of genus at least two, Hubbard and Masur proved
that the space of holomorphic quadratic differentials Q(S) on S is in a one to one correspondence with the space of measured foliations on S by associating with each holomorphic quadratic differential its corresponding vertical foliation. Thurston proved that the space of measured foliation on S is in a one to one correspondence with the space of measured geodesic laminations ML(S). Therefore, there is a bijection between Q(S) and ML(S) obtained by straightening the horizontal leaves of the quadratic differentials into hyperbolic geodesics on S and pushing forward the transverse measure.

Let X be an infinite Riemann surface whose covering group is of the first kind. We give an analogue of the Hubbard and Masur theorem for integrable holomorphic quadratic differentials on X. Namely, we give a complete characterization of the class of measured geodesic laminations on X that correspond to integrable holomorphic quadratic differentials thus establishing a bijection between the two spaces.

Applications of the above result:

Brownian motion on an arbitrary Riemann surface is recurrent if and only if almost every leaf of any integrable holomorphic quadratic differential is recurrent. If the Brownian motion on a Riemann surface is recurrent, then each integrable holomorphic quadratic differential can be approximated by a sequence of Jenkins-Strebel differentials in the L^1-norm. The Teichmuller metric on the Teichmuller space of a Riemann surface with recurrent Brownian motion is equal to the supremum of ½ times the logarithm of the ratio of the extremal lengths of simple closed curves (extension of Kerckhoff's formula for compact surfaces). An extension of McMullen's result for the Cantor tree surface to give sufficient conditions on the lengths of cuffs of a pants decomposition to guarantee that the Brownian motion is not recurrent. Speaker(s): Dragomir Saric (Queen's College CUNY)