A Kleinian group version of Torelli’s Theorem

Each closed Riemann surface S of genus g ≥ 1 has associated a principally polarized Abelian variety J(S), called the Jacobian variety of S. Classical Torelli’s theorem states that S is uniquely determined, up to conformal equivalence, by J(S). On the other hand, if S is either a non-compact analytically finite Riemann surfaces or an analytically finite Riemann orbifold, then it seems that there is not a natural way to associate to it a principal polarized Abelian variety. We survey some results concerning a Torelli’s type of theorem for the case of homology Riemann orbifolds and Kleinian groups.