The theorem mentioned in the title states:
Suppose $M$ and $N$ are complete finite-volume hyperbolic manifolds of dimension $n \ge 3$. If there exists an isomorphism $f:\pi_1(M) \rightarrow \pi_1(N)$ then it is induced by a unique isometry from $M$ to $N$.
So the first thing that comes in mind is: Why does it not work for dimension 2? I suppose it has to do with Gauß-Bonnet but I quite don't get why..
And I also read a lot that this theorem implies that geometric invariants are also topological invariants but they never explain why..
So my question is: Why? Is there an "easy" way to taste the fruit of wisdom to these questions?
The issue with genus-$g$ surfaces is that there is a $(6g-6)$-dimensional moduli space of complete finite-volume hyperbolic structures, where $g\geq 2$. These are parameterized by Fenchel-Nielsen coordinates.
Here is a simple example: given a non-separating geodesic simple closed curve, you can continuously modify the hyperbolic structure to shrink the geodesic's length. The identity map on the underlying topological space induces an isomorphism of fundamental groups, but there is no corresponding isometry. (More specifically, I mean you can change the length of one of the curves in a pants decomposition, which is one of the $6g-6$ coordinates.)