Dehn surgeries, group actions and Seifert fiber spaces

摘要

Let K be a knot in the 3-sphere S~3 and γ a slope (the isotopy class of an essential simple closed curve) on the boundary of the exterior of K. Then the manifold obtained from S~3 by γ-Dehn surgery on K is denoted by (K; γ). As usual, using the preferred meridian-longitude pair of K is contained in S~3, we parametrize slopes, and hence surgeries on K by r ∈ Q ∪ {∞}; we also write (K; r) for (K; γ). Suppose that K is hyperbolic, i.e., the complement S~3 - K admits a complete hyperbolic metric of finite volume.