Dragomir Saric

An infinite Riemann surface is parabolic if it does not support Green’s function. Hopf-Tsuji-Sullivan theorem states that X is parabolic iff the geodesic flow is ergodic iff the Brownian motion on X is recurrent iff the covering Fuchsian group is of divergence type. We consider the question of determining whether an explicitly given Riemann surface (in terms of Fenchel-Nielsen parameters) is parabolic. Some sufficient conditions are obtained depending on the size of the lengths and choice of relative twists. We also solve a conjecture of Kahn and Markovic by showing that lengths can be arbitrarily large while the surface remains parabolic when the twists are appropriately chosen.