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Homotopy hyperbolic 3-manifolds are hyperbolic

Gabai, David and Meyerhoff, G. Robert and Thurston, Nathaniel (2003) Homotopy hyperbolic 3-manifolds are hyperbolic. Annals of Mathematics, 157 (2). pp. 335-431. ISSN 0003-486X.

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This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold.

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Additional Information:(Received September 6, 1996); (Revised September 29, 2000) We thank The Geometry Center and especially Al Marden and David Epstein for the vital and multifaceted roles they played in this work. We also thank the Boston College Physics Department for allowing us to use their suite of computers. Jeff Weeks and SnapPea provided valuable data and ideas. In fact, the data from an undistributed version of SnapPea encouraged us to pursue a computer-assisted proof of Theorem 0.2. Bob Riley specially tailored his program Poincar´e to directly address the needs of our project. His work provided many leads in our search for killerwords. Further, he provided the first proof to show (experimentally) that the six exceptional regions (other than the Vol3 region) correspond to closed orientable 3-manifolds. The authors are deeply grateful for his help. The first-named author thanks the NSF for partial support. Some of the first author’s preliminary ideas were formulated while visiting David Epstein at the University of Warwick Mathematics Institute. The second-named author thanks the NSF and Boston College for partial support; the USC and Caltech Mathematics Departments for supporting him as a visitor while much of this work was done; and Jeff Weeks, Alan Meyerhoff, and especially Rob Gross for computer assistance. The third-named author thanks the NSF for partial support, and the Geometry Center and the Berkeley Mathematics Department for their support. Finally, we thank the referees for the magnificent job they did. The first set of referees read our paper thoroughly and made numerous excellent suggestions for improving the exposition. Further, their discussion of issues related to computer-aided proofs crystallized many of these topics in our minds. The second set of referees also read the paper thoroughly, and we are grateful for their elegant suggestions concerning the exposition. They also checked the programs in great detail, and approached this task with a desire to understand what was really going on behind the scenes. Their ingenious robustness checks raise the confidence level in our proof, and their thought-provoking comments should help us when we attempt to use the computer to help us push across the frontier of our current results.
Issue or Number:2
Record Number:CaltechAUTHORS:GABaom03
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ID Code:767
Deposited By: Archive Administrator
Deposited On:29 Sep 2005
Last Modified:02 Oct 2019 22:36

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