Manifolds and cell complexes Three-manifolds Theorem

Hyperbolic Dehn surgery theorem

Builds on Hyperbolic 3-manifold Dehn surgery Figure-eight knot

Statement

Let $M$ be an orientable hyperbolic 3-manifold of finite volume that is the interior of a compact $3$ -manifold $\bar{M}$ whose boundary consists of tori $T_1, \ldots, T_n$ . A slope on $T_i$ is an isotopy class of essential simple closed curves on $T_i$ . For slopes $s_1, \ldots, s_n$ , let $M(s_1, \ldots, s_n)$ be the closed manifold obtained by Dehn filling. It is made by gluing a solid torus to each $T_i$ so that a meridian is glued to $s_i$ . Then for each $i$ there is a finite set $E_i$ of slopes on $T_i$ such that $M(s_1, \ldots, s_n)$ is hyperbolic whenever $s_i \notin E_i$ for every $i$ .

Remarks

The theorem is due to Thurston. The complete structure on $M$ lies in a space of incomplete hyperbolic structures, parametrised near it by the shapes of ideal tetrahedra. Nearby incomplete structures whose metric completion is a closed manifold give the filled manifolds. Thurston also showed that the volumes of the filled manifolds are smaller than the volume of $M$ and tend to it as all slopes go to infinity. For the complement of the figure-eight knot, exactly ten slopes give a non-hyperbolic filling: $\infty$ , $0$ , $\pm 1$ , $\pm 2$ , $\pm 3$ and $\pm 4$ . The finite set cannot be empty in general: the filling along the meridian of a knot complement gives back $S^3$ , which is not hyperbolic.

Going deeper

For a student

The theorem produces infinitely many closed hyperbolic $3$ -manifolds from each cusped one, for example from knot complements. Since every closed orientable $3$ -manifold is obtained by surgery on a link in $S^3$ , it suggests that hyperbolic manifolds are generic. The exceptional slopes are well studied: Lackenby and Meyerhoff proved that a one-cusped manifold has at most ten of them, so the figure-eight knot complement is extremal.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Dehn surgery concept
Figure-eight knot example
Hyperbolic 3-manifold concept
Hyperbolic Dehn surgery theorem statement

Tag	`BB28P2ED5H`
URI	`m2://BB28P2ED5H`
Kind	statement, theorem
Writing	W0 stub
MSC	57K32, 57K30
Status	This copy of the site has no build data. Run `m2 status BB28P2ED5H`.
About	Hyperbolic 3-manifold, Dehn surgery
Uses	Hyperbolic 3-manifold, Dehn surgery, Figure-eight knot
Changed	2 November 2026