Finiteness of purely cosmetic fillings

Abstract A pair of Dehn fillings on a compact, orientable 3-manifold M with a torus boundary $\partial M$ is said to be purely cosmetic if the resulting 3-manifolds are orientation-preservingly homeomorphic. In this article, we show that if $\partial M$ is incompressible, then there are only finitely many pairs of purely cosmetic fillings.