Dehn Fillings of Large Hyperbolic 3-Manifolds

Abstract

Let $M$ be a compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus and which contains an essential closed surface $S$. It is conjectured that 5 is an upper bound for the distance between two slopes on $\partial M$ whose associated fillings are not hyperbolic manifolds. In this paper we verify the conjecture when the first Betti number of $M$ is at least 2 by showing that given a pseudo-Anosov mapping class $f$ of a surface and an essential simple closed curve $\gamma$ in the surface, then 5 is an upper bound for the diameter of the set of integers $n$ for which the composition of $f$ with the $n^{th}$ power of a Dehn twist along $\gamma$ is not pseudo-Anosov. This sharpens an inequality of Albert Fathi. For large manifolds $M$ of first Betti number 1 we obtain partial results. Set $$ \mathcal C(S) = {\rm{slopes} \, r \, | \, ker(\pi_1 (S) \to \pi_1 (M(r))) \neq {1} } $$ A singular slope for $S$ is a slope $r_0 \in \mathcal{C} (S)$ such that any other slope in $\mathcal{C}S$ is at most distance 1 from $r_0$. We prove that the distance between two exceptional filling slopes is at most 5 if either (i) there is a closed essential surface $S$ with $ \mathcal{C}S$ finite, or (ii) there are singular slopes $r_1 \neq r_2$ for closed essential surfaces $S_1, S_2$ in $M$.