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A method to prove that a manifold has a complete hyperbolic structure.
by
Harriet Moser
Columbia University
Let M be a finite volume manifold where the boundary of the closure consists of a disjoint union of a finite set of tori. Then M will have a complete hyperbolic structure if there exists a simultaneous solution to the consistency and completeness equations associated with some triangulation of M. Using Newton's Method, the computer program SnapPea can approximate if this solution exists. The talk will explore a method that uses the approximation in a test that can conclusively prove that the manifold has a complete hyperbolic structure. From the consistency and completeness equations we get a function f:Cn ® Cn, and we want to know if there is a solution to f(z)=0 in the upper half plane of Cn. We use the Kantorovich Theorem, which supplies the conditions for this solution to exist in a small neighborhhod near but not including a, the approximate solution given by SnapPea. This involves finding the Lipschitz Ratio, L, for the derivative of f in this neighborhood and the supremum norm of the matrix f¢(a)−1. Implementaion and examples, including the SnapPea cusped census, will be discussed.
Date received: February 8, 2005
Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capo-27.