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Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected hyperbolic 3-manifold. Let $X$ be the set of fixed points in $\overline{\mathbb{H}^3}$ of elements in $\rho (\pi_1(M))$ (all of which are isometries of loxodromic type). The closure of $X$ coincides with the limit set of $\Gamma$, which is a perfect set (=no isolated points).

What do we know about the topological properties of $X$ (or of the limit set) as a subspace of $\partial \mathbb{H}^3$? for instance, is it dense?

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$X$ is dense in the 2-sphere boundary $S^2_\infty = \partial\mathbb{H}^3$, and the limit set equals $S^2_\infty$.

An equivalent way of saying this is that closed orbits of the geodesic flow are dense in the unit tangent bundle of $\pi_1(M)$, and this is closely connected to E. Hopf's theorem that the geodesic flow is ergodic.

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