## Journal of Differential Geometry

- J. Differential Geom.
- Volume 58, Number 2 (2001), 263-308.

### Dehn Fillings of Large Hyperbolic 3-Manifolds

S. Boyer, C. McA. Gordon, and X. Zhang

#### 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 $M$.

#### Article information

**Source**

J. Differential Geom., Volume 58, Number 2 (2001), 263-308.

**Dates**

First available in Project Euclid: 20 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1090348327

**Digital Object Identifier**

doi:10.4310/jdg/1090348327

**Mathematical Reviews number (MathSciNet)**

MR1913944

**Zentralblatt MATH identifier**

1042.57007

#### Citation

Boyer, S.; Gordon, C. McA.; Zhang, X. Dehn Fillings of Large Hyperbolic 3-Manifolds. J. Differential Geom. 58 (2001), no. 2, 263--308. doi:10.4310/jdg/1090348327. https://projecteuclid.org/euclid.jdg/1090348327