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Let M be a 3–manifold whose boundary consists of tori. The computer program
SnapPea, created by Jeff Weeks, can approximate whether or not M is a complete
hyperbolic manifold. However, until now, there has been no way to determine
from this approximation if M is truly hyperbolic and complete. This paper
provides a method for proving that a manifold has a complete hyperbolic
structure based on the approximations of Snap, a program that includes the
functionality of SnapPea plus other features. The approximation is done by
triangulating M, identifying consistency and completeness equations as described by
Neumann and Zagier [Topology 24 (1985) 307–332] and Benedetti and Petronio
[Lectures on hyperbolic geometry, Universitext, Springer, Berlin (1992)] with
respect to this triangulation, and then, according to Weeks [”Handbook of
Knot Theory”, Elsevier, Amsterdam (2005) 461–480], trying to solve the
system of equations using Newton’s Method. This produces an approximate,
not actual solution. The method here uses the Kantorovich Theorem to
prove that an actual solution exists, thereby assuring that the manifold has a
complete hyperbolic structure. Using this, we can definitively prove that
every manifold in the SnapPea cusped census has a complete hyperbolic
structure.
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