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Thurston's hyperbolic Dehn filling theorem for finitely many cusps

Statement

Assume the Axiom of Choice. Let $M$ be the interior of a compact connected orientable three-manifold with $k \geq 1$ torus cusps and a complete finite-volume hyperbolic metric. Choose an ordered peripheral basis on each cusp. For cusp $i$ , let the generalized filling parameter sphere be $S_{i} = R^{2} \cup {\infty}$ , with $\infty$ denoting the complete cusp and primitive integral pairs modulo sign denoting ordinary slopes. There is a neighborhood $U$ of $(\infty, \dots, \infty)$ in $\prod_{i = 1}^{k} S_{i}$ such that every tuple in $U$ whose finite coordinates are primitive integral slopes determines a complete finite-volume hyperbolic metric on the corresponding ordinary Dehn filling; coordinates equal to $\infty$ remain unfilled complete cusps. Consequently, one may choose neighborhoods $U_{i}$ of $\infty$ with $\prod_{i} U_{i} \subseteq U$ , and only finitely many primitive integral pairs at each cusp lie outside $U_{i}$ . Thus every simultaneous filling whose slopes all lie in their respective $U_{i}$ is hyperbolic. Nonprimitive ray parameters may describe cone or orbifold completions but are not ordinary smooth fillings. No universal numeric length threshold or universal bound on the number of exceptional slopes is asserted.

Statement references

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 5, §5.8, Theorem 5.8.2

Petronio and Porti, Negatively Oriented Ideal Triangulations and a Proof of Thurston’s Hyperbolic Dehn Filling Theorem, Theorem 0.1

Facts & Assumptions

Given: AC, the complete finite-volume metric with $k \geq 1$ torus cusps and the chosen ordered peripheral bases.

[A1]

AC and its countable-choice consequence are retained. The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), AC implies DC implies countable choice

[F1]

The full $k$ -cusped local deformation-and-completeness theorem gives a product neighborhood of the complete generalized parameters and smooth complete finite-volume ordinary fillings for every primitive tuple therein, with the other cusps unfilled. Peripheral holonomy equations produce smooth hyperbolic fillings

[F2]

Slopes and arbitrary supplied ordinary gluing maps have the stated primitive-pair convention. Slopes and generalized fillings on torus cusps

[F3]

The input metric and volume convention are Pared hyperbolic structures and volume. Each chosen-section cusp length is a norm on its rank-two peripheral lattice; changing section rescales it by a positive constant. Euclidean length of a slope on a horospherical cusp section

Proof

1.1A1F1F2F3given

Apply [F1] to the entire given $k$ -cusped metric. It supplies a product neighborhood $U$ of $(\infty, \dots, \infty)$ and the required complete finite-volume metric for every tuple with primitive finite coordinates. Coordinates $\infty$ retain their complete translation cusps, so the conclusion includes every partial filling and the unchanged complete manifold. These are the ordinary fillings of [F2], including their supplied gluing data, rather than just representations solving the peripheral equations.

2.1F1F2step 1.1constructalgebra

Choose cuspwise open neighborhoods $U_{i}$ of $\infty$ with $\prod_{i} U_{i} \subseteq U$ ; one may use the product neighborhood in [F1] itself. The complement $S_{i} ∖ U_{i}$ is a closed subset of the compact sphere $S_{i}$ missing $\infty$ , hence a compact subset of $R^{2}$ and therefore bounded. A bounded set contains only finitely many integer lattice points: if its norm is at most $B$ , each coordinate is one of the finitely many integers between $- ⌈ B ⌉$ and $⌈ B ⌉$ . Restricting to primitive pairs and quotienting by simultaneous sign cannot increase their number. This proves the finite-exception assertion at every cusp and hence the simultaneous conclusion.

3.1F2F3step 2.1algebra∎

By [F3], any sequence of primitive slopes whose chosen-section lengths tend to infinity leaves every bounded coefficient set, so it eventually lies in its $U_{i}$ . Rescaling a horotorus multiplies all these lengths by one positive constant and does not alter this conclusion. A nonprimitive ray parameter does not specify an ordinary primitive meridian; by [F2] its cone-angle convention is separate from the smooth fillings proved here. All neighborhoods are allowed to depend on the given manifold and its metric, so the proof asserts neither a universal numeric threshold nor a universal number of exceptions.

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Sources

William P. Thurston, The Geometry and Topology of Three-Manifolds (standard reference, not scraped)
Carlo Petronio and Joan Porti, Negatively Oriented Ideal Triangulations and a Proof of Thurston’s Hyperbolic Dehn Filling Theorem (standard reference, not scraped)