Abstract: The main result proved in this thesis is the existence
of a complete finite volume Riemannian metric in the interior of any nontrivial
knot complement in the 3-sphere. Also this result is generalised to any
irreducible subcomplex complement in the 3-sphere with respect to some
triangulation of the sphere. A theorem of Thurston says that a knot complement
in the 3-sphere supports hyperbolic metric if and only if the knot is neither
a torus knot nor a satellite knot. In the metric sense the above result
is a generalization of this theorem of Thurston. Using this existence of
nonpositively curved metric in a knot complement and by Farrell and Jones
Topological Rigidity theorem we prove a conjecture of S. Cappell. The conjecture
says that the surgery groups of the fundamental group of any knot complement
is isomorphic to the surgery groups of the infinite cyclic group. Using
the similar technique we compute explicitely the surgery groups of any
subcomplex complement in the 3-sphere with respect to some triangulation
of the sphere.
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