The maximal number of exceptional Dehn surgeries

Abstract

Thurston’s hyperbolic Dehn surgery theorem is one of the most important results in 3-manifold theory, and it has stimulated an enormous amount of research. If M is a compact orientable hyperbolic 3-manifold with boundary a single torus, then the theorem asserts that, for all but finitely many slopes s on ∂M , the manifold M(s) obtained by Dehn filling along s also admits a hyperbolic structure. The slopes s where M(s) is not hyperbolic are known as exceptional. A major open question has been: what is the maximal number of exceptional slopes on such a manifold M? When M is the exterior of the figure-eight knot, the number of exceptional slopes is 10, and this was conjectured by Gordon in [19] to be an upper bound that holds for all M . In this paper, we prove this conjecture.