Dehn Surgery and Negatively Curved 3-manifolds

Dehn surgery is perhaps the most common way of constructing 3-manifolds, and yet there remain some profound mysteries about its behaviour. For example, it is still not known whether there exists a 3-manifold which can be obtained from S by surgery along an infinite number of distinct knots. 1 (See Problem 3.6 (D) of Kirby’s list [9]). In this paper, we offer a partial solution to this problem, and exhibit many new results about Dehn surgery. The methods we employ make use of well-known constructions of negatively curved metrics on certain 3-manifolds.