Kleinian group version of Torelli's theorem (Q2907940)

Each closed Riemann surface \(S\) of genus \(g \geq 1\) has an associated principally polarized Abelian variety \(J (S)\), called the Jacobian variety of \(S\). The classical Torelli 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 no natural way to associate to it a principal polarized Abelian variety. In this paper, the author surveys some results concerning Torelli type theorems for the case of homology Riemann orbifolds, that is the Riemann orbifolds with the property that the derived subgroup of their orbifold fundamental group uniformizes a closed Riemann surface, also in terms of Kleinian groups.