Geometry and Topology Seminar

Let M be a 1-cusped hyperbolic 3-manifold whose cusp shape is not quadratic, and M(p/q) be its p/q-Dehn filled manifold. If p/q is not equal to p'/q' for sufficiently large |p|+|q| and |p'|+|q'|, there is no orientation preserving isometry between M(p/q) and M(p'/q'). This resolves the conjecture of C. Gordon, which is so called the Cosmetic Surgery Conjecture, for hyperbolic 3-manifolds belonging to the aforementioned class except for possibly finitely many exceptions for each manifold. We also consider its generalization to more cusped manifolds. The key ingredient of the proof is the unlikely intersection theory developed by E. Bombieri, D. Masser, and U. Zannier.