Algebraic & Geometric Topology

Proving a manifold to be hyperbolic once it has been approximated to be so

Harriet Moser

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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.

Article information

Algebr. Geom. Topol., Volume 9, Number 1 (2009), 103-133.

Received: 14 August 2008
Accepted: 15 September 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 57N16: Geometric structures on manifolds [See also 57M50]
Secondary: 54E50: Complete metric spaces 51H20: Topological geometries on manifolds [See also 57-XX]

3–manifold hyperbolic complete approximate


Moser, Harriet. Proving a manifold to be hyperbolic once it has been approximated to be so. Algebr. Geom. Topol. 9 (2009), no. 1, 103--133. doi:10.2140/agt.2009.9.103.

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