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In some specific situation there are some nice relations between a Kleinian group and its quotient manifold. For example, if $G$ is a once-punctured-torus group (i.e. a free subgroup of $\mathbb{P}SL(2,\mathbb{C})$ on two generators such that the commutator of the generators is parabolic) then the quotient manifold $\mathbb{H}^3/G$ is homeomorphic to $T_1\times (-1,1)$, where $T_1$ is the once-punctured torus. Nevertheless, this is not enough to determine the (conjugacy class) of $G$ in $\mathbb{P}SL(2,\mathbb{C})$, for we need a pair of invariants (the end invariants) to solve this problem, which have a nice description as well.

I was wondering whether there exist similar relations for different cases. For instance, consider a Kleinian group which is the image of the fundamental group of a compact orientable surface of genus $g$ ($g>1$) under a faithful representation $$\rho\colon\pi_1(S_g)\to \mathbb{P}SL(2,\mathbb{C})$$ What can we say about the quotient $\mathbb{H}^3/\rho(\pi_1(S_g))$?

Moreover, it would be useful to have some result in the opposite direction too, that is obtaining information about the group starting from the quotient.

Thank you.

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