ERGODICITY OF THE GEODESIC FLOW ON SYMMETRIC SURFACES

. We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that guarantee the surface 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. When the surface X is end symmetric, we prove that 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 such that 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].