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.