On hyperbolic 3–manifolds realizing the maximal distance between toroidal Dehn fillings

Abstract

For a hyperbolic 3-manifold M with a torus boundary component, all but finitely many Dehn fillings on the torus component yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where M has two exceptional Dehn fillings, both of which yield toroidal manifolds. For such situation, Gordon gave an upper bound for the distance between two slopes of Dehn fillings. In particular, if M is large, then the distance is at most 5. We show that this upper bound can be improved by 1 for a broad class of large manifolds.

Toroidal Dehn fillings on large hyperbolic 3-manifolds

“…Suppose Δ(α, β) = 5. Then, we showed that ∂M is a single torus or two tori [6]. By [14], the latter case happens only when M is the Whitehead sister link exterior.…”

“…If not, Δ(α, β) = 5 [7]. By [6], ∂M is a single torus or two tori. However, the latter case does not happen by [14].…”

“…As shown in [6, Proposition 2.3], we may assume that at least one essential torus is separating. By following the arguments of [6], we will see that this torus meets the core of the attached solid torus only twice. The argument is divided into three cases, according to how many times the other essential torus meets the core of the attached solid torus.…”

“…Assume that the two vertices of G T have distinct signs. Then, there is only one possible graph pair as shown in Figure 7 by [6,Lemmas 7.3 and 7.4]. Note that the jumping number is two.…”

“…By [6,Proposition 8.7], there are only two possibilities for G P : H(3, 1, 1) and H (3, 2, 0). If G P = H (3, 1, 1), then G T = G (1, 1, 1, 1, 0).…”