What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

Question

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for the action of $\Gamma$ on $V$?

What can be said about orbits of this action?

I have added the ergodic theory label because maybe people expert in ergodic theory may provide good answers!

@ Glougloubarbaki

for a closed hyperbolic manifold the set $\Omega$ should be always empty..

Answer 1

The action of $\Gamma$ on $S^1$ is ergodic with respect to the Lebesgue measure class on $S^1$, and the set of points fixed by elements of $\Gamma$ is countable. So no, there is no good notion of a fundamental domain for the action on $V$.

Answer 2

The interesting question is the action on the largest open subset of the reals on which $\Gamma$ acts properly discontinuously (denote it by $\Omega$). Then $\mathbb{H} \cup \Omega/\Gamma$ is a complex Riemann surface with boundary.

$\Omega/\Gamma$ is called the ideal boundary of $S$. It may be empty. It plays an important role in Teichmüller theory, see e.g. the book by Gardiner and Lakic, "Quasiconformal Teichmüller theory".

In particular, whenever $\Omega \neq \mathbb R$, there is no nice fundamental domain on $\mathbb R$ for the action of $\Gamma$.