Dragomir Saric, from CUNY Graduate Center and Queens College, will present a colloquium titled “The Finite-Area Holomorphic Quadratic Differentials and the Geodesic Flow on Infinite Riemann Surface” as part of the Mathematics Department Colloquium Lecture series.

Abstract: Let  be an infinite Riemann surface with a conformally hyperbolic metric. The Hopf-Tsuji-Sullivan theorem states that the geodesic flow is ergodic iff the Poincare series is divergent iff the Brownian motion is recurrent, and many other equivalent conditions are given in the literature. We added an equivalent condition: the Brownian motion on  is recurrent iff almost every horizontal leaf of every finite-area holomorphic quadratic differential is recurrent.

A finite-area holomorphic quadratic differential on  is uniquely determined by the homotopy class of its horizontal foliation, uniquely represented by a measured geodesic lamination on . Most measured geodesic laminations do not come from the horizontal foliations of finite-area differentials. The problem of intrinsically deciding which measured laminations are induced by finite-area differentials is highly transcendental. From now on, assume that  is equipped with a geodesic pants decomposition whose cuffs are bounded. The space of finite-area holomorphic quadratic differentials on $X is in a one-to-one correspondence with the measured geodesic laminations on  whose intersection numbers with the cuffs (and “adjoint cuffs”) are square summable. Using this parametrization, we establish that the Brownian motion on  is recurrent iff the simple random walk on the graph dual to the pants decomposition is recurrent.