I'm not qualified to certify optimality, but I've always thought that the Mostow rigidity theorem is a good candidate. The theorem says that every isomorphism between the fundamental groups of two finite volume hyperbolic manifolds of dimension at least 3 is induced by a unique isometry. Mostow's original proof (for the compact case) used: - Riemannian geometry - Conformal geometry - Geometric group theory - Representation theory - Ergodic theory - A dash of number theory For generalizations to symmetric spaces you need algebraic geometry and more serious number theory as well.