The geodesic flow on infinite Riemann surfaces has been studied for a century. In 1939, E. Hopf proved that if almost every geodesic on a surface X is recurrent, then the geodesic flow on the unit tangent bundle T^1X is ergodic; and if there is a positive measure set of transient geodesics on X, then almost every geodesic is transient. A theorem of Hopf-Tsuji-Sullivan states that the geodesic flow is ergodic iff the Poincare series (of the covering Fuchsian group) diverges iff the Brownian motion is recurrent. Ahlfors-Sario proved that the Brownian motion on X is recurrent iff X does not support a Green’s function, in notation, X in O_G. There are many other equivalent conditions to X in O_G.
 
The type problem asks when an explicitly constructed Riemann X is in O_G (i.e., the geodesic flow is ergodic). Classically, the constructions are given gluing slit planes depending on some sets of parameters or constructions of Riemann surfaces as infinite (branched) coverings of compact surfaces. We are interested in a general situation of arbitrary Riemann surfaces and deciding (as much as possible) whether they are in O_G from the Fenchel-Nielsen parameters.
 
In a joint work with A. Basmajian and H. Hakobyan, we find sufficient conditions on the Fenchel-Nielsen parameters such that X is in O_G. It turns out that the conditions largely depend on the size of the space of (topological) ends of X. In the case of a Cantor set of ends, if the length parameters (of the Fenchel-Nielsen parametrization) go to zero at a rate n/2^n on the n-th level, then the surface X is in O_G. McMullen proved that if the lengths at the level n are bounded between two positive constants, independent of n, then the surface X is not in O_G. M. Pandazis proved that if the rate of the length parameters is n^r/2^n for any r>1, then X is in O_G thus closing the gap. More importantly, the ergodicity of the geodesic flow is not dependent on the twist parameters (of the Fenchel-Nielsen parametrization) in the case of the Cantor set of ends.
 
From now on, X has finitely many topological ends or, at most, countably many. Together with Basmajian and Hakobyan, we proved that X is in O_G if the lengths are at most 2*log(n) with arbitrary twists. Thus, lengths can go to infinity at a mild rate, and the geodesic flow stays ergodic. We also proved that the twists have an influence; for example, if all twists are ½, then the lengths can be 4*log(n), and still, X is in O_G.
 
One could expect that larger rates of the length parameters going to infinity would force the geodesic flow to be non-ergodic because of Hopf’s theorem. Contrary to this and motivated by their proof of the Surface Subgroup Theorem, J. Kahn and V. Markovic conjectured that, given arbitrary large length parameters, there is always a choice of twists such that the resulting surface X is in O_G. In joint work with Hakobyan and Pandazis, we establish the validity of this conjecture if one allows increasing lengths even further and chooses all twists to be exactly ½.

For more information check https://www.groups-and-spaces.kit.edu/115_808.php