The '''Thurston's hyperbolic Dehn surgery theorem'
states M(u_1, u_2, \dots, u_n) is hyperbolic as long as a finite set of exceptional slopes
E_i is avoided for the i
-th cusp for each i
. M(u_1, u_2, \dots, u_n) converges to M
in H
as all p_i^2+q_i^2 \rightarrow \infty for all p_i/q_i corresponding to non-empty Dehn fillings 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. 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 \omega^\omega. This result is known as the
Thurston-Jørgensen theorem. Further work characterizing this set was done by
Gromov. The
figure-eight knot and the
(-2, 3, 7) pretzel knot are the only two knots whose complements are known to have more than 6 exceptional surgeries; they have 10 and 7, respectively.
Cameron Gordon conjectured that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot complement. This was proved by Marc Lackenby and Rob Meyerhoff, who show that the number of exceptional slopes is 10 for any compact orientable 3-manifold with boundary a torus and interior finite-volume hyperbolic. Their proof relies on the proof of the
geometrization conjecture originated by
Grigori Perelman and on
computer assistance. It is currently unknown whether the figure-eight knot is the only one that achieves the bound of 10. One conjecture is that the bound (except for the two knots mentioned) is 6. Agol has shown that there are only finitely many cases in which the number of exceptional slopes is 9 or 10. ==References==