A Kleinian group version of Torelli’s Theorem

Abstract

Each closed Riemann surface \(S\) of genus \(g \geq 1\) 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.