Ergodicity of action of finite index subgroups in the boundary

Question

Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}$. Does it follow that any finite index subgroup $\Gamma_0$ of $\Gamma$ also acts ergodically on $\partial{\mathbb{H}^2}$?

Via the (Poisson boundary) correspondence of harmonic functions with $L^{\infty}$ functions on the boundary this is equivalent to the following: Suppose all bounded harmonic functions of a hyperbolic surface $S$ are constant, does the same follow for finite covers of $S$?

More generally, I am interested in the following question: Let $G$ be a real semi-simple Lie group of finite center and let $\Gamma$ be a discrete subgroup of $G$, Let $P$ be the minimal parabolic subgroup (or possibly any other closed subgroup). If $\Gamma$ acts ergodically on $G/P$, do finite index subgroups of $\Gamma$ also act ergodically on $G/P$?

Any results related to this question are also very welcome.

Thanks.

Answer 1

Let $X$ be a Riemann surface of class $P_G$ (i.e. which carries a Green function) but is Liouville (i.e. admits no nonconstant bounded harmonic functions). One way to construct these is to take a $\mathbb Z^3$-covering of a compact hyperbolic Riemann surface, see

Lyons, Terry; Sullivan, Dennis, Function theory, random paths and covering spaces, J. Differ. Geom. 19, 299-323 (1984). ZBL0554.58022.

On the other hand, every hyperbolic surface $X$ of class $P_G$ has a finite cover $Y\to X$ such that $Y$ is of class $P_{HB}$, i.e. admits nonconstant bounded harmonic functions, see

Uludağ, A. Muhammed, Existence of Green function and bounded harmonic functions on Galois covers of Riemannian manifolds, Osaka J. Math. 38, No. 2, 295-301 (2001). ZBL0994.31002.

This gives you examples of Fuchsian groups acting ergodically on $S^1$ and containing finite index subgroups whose actions on $S^1$ are not ergodic.