Reducible And Finite Dehn Fillings

We show that the distance between a finite filling slope and a reducible filling slope on the boundary of a hyperbolic knot manifold is at most one. Let M be a knot manifold, i.e. a connected, compact, orientable 3-manifold whose boundary is a torus. A knot manifold is said to be hyperbolic if its interior admits a complete hyperbolic metric of finite volume. Let M(α) denote the manifold obtained by Dehn filling M with slope α and let ∆(α, β) denote the distance between two slopes α and β on ∂M . When M is hyperbolic but M(α) isn’t, we call the corresponding filling (slope) an exceptional filling (slope). Perelman’s recent proof of Thurston’s geometrisation conjecture implies that a filling is exceptional if and only if it is either reducible, toroidal, or Seifert fibred. These include all manifolds whose fundamental groups are either cyclic, finite, or very small (i.e. contain no non-abelian free subgroup). Sharp upper bounds on the distance between exceptional filling slopes of various types have been established in many cases, including: • ∆(α, β) ≤ 1 if both α and β are reducible filling slopes [GL2] • ∆(α, β) ≤ 1 if both α and β are cyclic filling slopes [CGLS] • ∆(α, β) ≤ 1 if α is a cyclic filling slope and β is a reducible filling slope [BZ2] • ∆(α, β) ≤ 2 if α is a cyclic filling slope and β is a finite filling slope [BZ1] • ∆(α, β) ≤ 2 if α is a reducible filling slope and β is a very small filling slope [BCSZ2] • ∆(α, β) ≤ 3 if both α and β are finite filling slopes [BZ3] • ∆(α, β) ≤ 3 if α is a reducible filling slope and β is a toroidal filling slope [Wu] [Oh] • ∆(α, β) ≤ 8 if both α and β are toroidal filling slopes [Go] ∗Partially supported by NSERC grant RGPIN 9446