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Peripheral holonomy equations produce smooth hyperbolic fillings

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 a peripheral basis $(μ_{i}, λ_{i})$ on each cusp. On the augmented holonomy space obtained by selecting one common invariant line on each peripheral subgroup, near the complete structure write $u_{i}$ and $v_{i}$ for the logarithms of the eigenvalue ratios of the peripheral holonomies of $μ_{i}$ and $λ_{i}$ , with branches chosen so $u_{i} = v_{i} = 0$ at the complete structure. There is a neighborhood of the complete point in the $k$ generalized parameter spheres such that primitive integral filling data $(p_{i}, q_{i})$ in that neighborhood satisfying $p_{i} u_{i} + q_{i} v_{i} = 2 π i$ at every filled cusp produce a complete finite-volume hyperbolic structure on the ordinary filled manifold. The peripheral full-turn equations are solved simultaneously across the $k$ cusps; unfilled cusps retain their complete parabolic holonomy. No universal numeric length threshold is asserted.

Statement references

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

Petronio and Porti, Negatively Oriented Ideal Triangulations and a Proof of Thurston’s Hyperbolic Dehn Filling Theorem, Theorem 0.1 and §§1–3

Facts & Assumptions

Given: AC, the full finite-volume $k$ -cusped metric, the ordered peripheral bases and the primitive filling data.

[A1]

AC supplies DC and countable choice. The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), AC implies DC implies countable choice

[F1]

The given metric has a Kleinian quotient description and a compact smooth core with standard torus cusps. Complete curvature minus one three-manifolds are Kleinian quotients, Finite-volume hyperbolic three-manifolds have finite standard cusp decompositions

[F2]

The augmented deformation germ is smooth of complex dimension $k$ , with independent logarithms $u_{i}$ and $v_{i} = u_{i} τ_{i} (u)$ , where the complete $τ_{i}$ are nonreal; for positive bases their imaginary parts are positive. Augmented holonomy coordinates at a complete finite-volume hyperbolic structure

[F3]

These factors admit a product neighborhood of the complete generalized parameters with unique simultaneous small full-turn solutions, continuous also at partially unfilled tuples. Simultaneous peripheral full-turn equations have unique small solutions

[F4]

Given such primitive solutions, the explicit developed ends extend to smooth solid-torus cores; the resulting metric is complete and has finite volume. Coordinates $u_{i} = 0$ remain complete cusps. Small augmented holonomy deformations have smooth geometric fillings

[F5]

Ordinary fillings use primitive slopes modulo sign, and cusp lengths use chosen-section Euclidean norms. Slopes and generalized fillings on torus cusps, Euclidean length of a slope on a horospherical cusp section, Pared hyperbolic structures and volume

Proof

1.1A1F1F2given

By [F1], the input metric satisfies the standard cusp and quotient hypotheses of [F2]. Temporarily orient every peripheral basis positively by replacing its second element by its negative exactly where necessary. The augmented theorem [F2] supplies one common neighborhood of $0 \in C^{k}$ with holomorphic $τ_{i}$ and $Im τ_{i} (0) > 0$ . Changing back the second basis element negates $v_{i}$ and $τ_{i}$ , so their complete values remain nonreal for every original basis. The logarithms are taken on this augmented space; their branched relationship with ordinary trace-squared coordinates is part of [F2].

2.1F2F3step 1.1construct

Shrink the small neighborhood enough for [F4]. Apply [F3] there to all $k$ factors simultaneously, rather than solving independent one-cusp deformations. It gives a radius $R$ and a product neighborhood $V = \prod_{i} ({(p, q) : p^{2} + q^{2} > R^{2}} \cup {\infty})$ in which every tuple has its unique small solution. Its $i$ th coordinate is zero exactly for the parameters $\infty$ . At every finite coordinate it satisfies $p_{i} u_{i} + q_{i} v_{i} = 2 π i$ . This is the asserted neighborhood and the full simultaneous solution.

3.1F4F5step 1.1step 2.1algebra∎

At a primitive integral coordinate, [F4] realizes the holonomy family by a metric and identifies the developed end as a punctured ordinary tube whose meridional angular period is precisely one full turn. It fills that end by a smooth geodesic core. Its complete proof treats the finite collection of ends together, proves metric completeness and finite volume, and leaves each $u_{i} = 0$ end a complete translation cusp. If a supplied basis was negative, change $q_{i}$ to $- q_{i}$ along with $v_{i}$ to $- v_{i}$ before applying that supplier; the equation and unoriented primitive slope are unchanged. Return afterward to the supplied bases. Thus the construction applies to every allowed tuple and its ordinary filling data. By [F5], no nonprimitive coefficient is being called an ordinary slope. The constants depend on the original metric and the cusp coordinates, as permitted; no universal length bound is asserted.

Depends on

Used by

Thurston's hyperbolic Dehn filling theorem for finitely many cusps Theorem

Dependency tree · two levels

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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)