Michael Pandazis

A Riemann surface equipped with its conformal hyperbolic metric is parabolic if and only if the geodesic flow on its unit tangent bundle is ergodic. Let X be a Cantor tree or a blooming Cantor tree Riemann surface. Fix a geodesic pants decomposition of X and call the boundary geodesics in the decomposition cuffs. Basmajian, Hakobyan, and \vSari\'c proved that if the lengths of cuffs are rapidly converging to zero, then X is parabolic. More recently, \vSari\'c proved a slightly slower convergence of lengths of cuffs to zero implies X is not parabolic. In the paper, we interpolate between the two rates of convergence of the cuffs to zero and find that these surfaces are not parabolic, thus completing the picture.

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X X that guarantee the surface X X is of parabolic type. An interesting class of Riemann surfaces for this problem is the one with finitely many topological ends. In this case the length part of the Fenchel-Nielsen coordinates can go to infinity for parabolic X X . When the surface X X is end symmetric, we prove that X X being parabolic is equivalent to the covering group being of the first kind. Then we give necessary and sufficient conditions on the Fenchel-Nielsen coordinates of a half-twist symmetric surface X X such that X X is parabolic. As an application, we solve an open question from the prior work of Basmajian, Hakobyan and the second author [Proc. Lond. Math. Soc. (3) 125 (2022), pp. 568–625].