conjecture (now proven) states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot. Cyclic surgery theorem The cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is not a Seifert-fibered space and r,s are slopes on T such that their Dehn fillings have cyclic fundamental group, then the distance between r and s (the minimal number of times that two simple closed curves in T representing r and s must intersect) is at most 1. Consequently, there are at most three Dehn fillings of M with cyclic fundamental group. Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem Thurston's hyperbolic Dehn surgery theorem states: is hyperbolic as long as a finite set of exceptional slopes is avoided for the i-th cusp for each i. In addition, converges to M in H as all for all corresponding to non-empty Dehn fillings . 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 . 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. Thurston's hyperbolization theorem for Haken manifolds One form of Thurston's geometrization theorem states: If M is an compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume. The Mostow rigidity theorem implies that if a manifold