3-manifold
Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem

Thurston's hyperbolic Dehn surgery theorem states: M (u 1, u 2, …, u n) {\displaystyle M(u_{1},u_{2},\dots,u_{n})} is hyperbolic as long as a finite set of exceptional slopes E i {\displaystyle E_{i}} is avoided for the i -th cusp for each i. In addition, M (u 1, u 2, …, u n) {\displaystyle M(u_{1},u_{2},\dots,u_{n})} converges to M in H as all p i 2 + q i 2 → ∞ {\displaystyle p_{i}^{2}+q_{i}^{2}\rightarrow \infty } for all p i / q i {\displaystyle p_{i}/q_{i}} corresponding to non-empty Dehn fillings u i {\displaystyle u_{i}}.

This theorem is due to William Thurston and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in H. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.

Another important result by Thurston is that volume decreases under hyperbolic Dehn filling. In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the Gromov norm.

Jørgensen also showed that the volume function on this space is a continuous, proper function. Thus by the previous results, nontrivial limits in H are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has ordinal type ω ω {\displaystyle \omega ^{\omega }}. This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov.

Also, Gabai, Meyerhoff & Milley showed that the Weeks manifold has the smallest volume of any closed orientable hyperbolic 3-manifold.
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