## Algebraic & Geometric Topology

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

Harriet Moser

#### Abstract

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

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

Dates
Accepted: 15 September 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796958

Digital Object Identifier
doi:10.2140/agt.2009.9.103

Mathematical Reviews number (MathSciNet)
MR2471132

Zentralblatt MATH identifier
1170.57015

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796958

#### References

• R Benedetti, C Petronio, Lectures on hyperbolic geometry, Universitext, Springer, Berlin (1992)
• Y-E Choi, Positively oriented ideal triangulations on hyperbolic three-manifolds, Topology 43 (2004) 1345–1371
• CPAN, Comprehensive Perl Archive Network Available at \setbox0\makeatletter\@url http://www.perl.com {\unhbox0
• C H Edwards, Jr, Advanced calculus of several variables, Dover Publications, New York (1994) Corrected reprint of the 1973 original
• D Gabai, R Meyerhoff, P Milley, Minimum volume cusped hyperbolic three-manifolds
• D Gabai, R Meyerhoff, P Milley, Mom technology and volumes of hyperbolic $3$–manifolds
• O Goodman, Snap Available at \setbox0\makeatletter\@url http:/www.ms.unimelb.edu.au//~snap \unhbox0
• J H Hubbard, B B Hubbard, Vector calculus, linear algebra, and differential forms. A unified approach, Prentice Hall, Upper Saddle River, NJ (1999)
• C J Leininger, Small curvature surfaces in hyperbolic $3$–manifolds, J. Knot Theory Ramifications 15 (2006) 379–411
• H H Moser, Proving a manifold to be hyperbolic once it has been approximated to be so, PhD thesis, Columbia University (2005) Available at \setbox0\makeatletter\@url http://www.math.columbia.edu/~moser {\unhbox0
• W D Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic $3$–manifolds, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 243–271
• W D Neumann, A W Reid, Arithmetic of hyperbolic manifolds, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 273–310
• W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307–332
• Pari-Gp, Computer algebra system Available at \setbox0\makeatletter\@url http://pari.math.u-bordeaux.fr/ {\unhbox0
• R M Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Math. 108, Springer, New York (1986)
• J G Ratcliffe, Foundations of hyperbolic manifolds, second edition, Graduate Texts in Math. 149, Springer, New York (2006)
• W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
• W P Thurston, Hyperbolic structures on $3$–manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. $(2)$ 124 (1986) 203–246
• W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Math. Series 35, Princeton Univ. Press (1997) Edited by S Levy
• J Weeks, SnapPea Available at \setbox0\makeatletter\@url http://www.northnet.org/weeks {\unhbox0
• J Weeks, Computation of hyperbolic structures in knot theory, from: “Handbook of knot theory”, (W Menasco, M Thistlethwaite, editors), Elsevier B. V., Amsterdam (2005) 461–480
• H Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont. (1972)