**Question 1:** Suppose that two hyperbolic 3-manifolds $M\_1$ and $M\_2$ with finitely generated fundamental group satisfy the property that for every closed geodesic in $M\_1$, there is a closed geodesic in $M\_2$ with the same length. Are $M\_1$ and $M\_2$ isometric?


**Question 2:** The same as question one, but suppose that we also know the corresponding geodesics have the same rotational holonomy.


We can rephrase these questions in the language of Kleinian groups, as whether the multipliers of elements of a Kleinian group determine the group.


For finite volume manifolds, it should suffice to prove that $M\_1$ and $M\_2$ have the same volume.



Does length spectrum determine a hyperbolic 3-manifold? What if we also know holonomies?
I think your questions are answered by [this](https://arxiv.org/abs/math/0606343) paper of Leininger, McReynolds, Neumann and Reid. In their terminology, your first question is asking about manifolds with equal *length sets*, and your second question is asking about manifolds with equal *complex length sets*. Their Theorem 1.3 constructs many pairs of hyperbolic 3-manifolds with equal length sets but different volumes, and in §6.2 they note that these manifolds actually have equal complex length sets.


