## Minimum volume cusped hyperbolic three-manifolds

HTML articles powered by AMS MathViewer

- by David Gabai, Robert Meyerhoff and Peter Milley PDF
- J. Amer. Math. Soc.
**22**(2009), 1157-1215

## Abstract:

This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic $3$-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic $3$-manifold with volume $\le 2.848$ can be obtained by a Dehn filling on one of $21$ cusped hyperbolic $3$-manifolds. We also show how this result can be used to construct a complete list of all one-cusped hyperbolic $3$-manifolds with volume $\le 2.848$ and all closed hyperbolic $3$-manifolds with volume $\le 0.943$. In particular, the Weeks manifold is the unique smallest volume closed orientable hyperbolic $3$-manifold.## References

- Colin C. Adams,
*The noncompact hyperbolic $3$-manifold of minimal volume*, Proc. Amer. Math. Soc.**100**(1987), no. 4, 601–606. MR**894423**, DOI 10.1090/S0002-9939-1987-0894423-8 - Ian Agol, Marc Culler, and Peter B. Shalen,
*Dehn surgery, homology and hyperbolic volume*, Algebr. Geom. Topol.**6**(2006), 2297–2312. MR**2286027**, DOI 10.2140/agt.2006.6.2297 - Ian Agol,
*Volume change under drilling*, Geom. Topol.**6**(2002), 905–916. MR**1943385**, DOI 10.2140/gt.2002.6.905 - Ian Agol, Peter A. Storm, and William P. Thurston,
*Lower bounds on volumes of hyperbolic Haken 3-manifolds*, J. Amer. Math. Soc.**20**(2007), no. 4, 1053–1077. With an appendix by Nathan Dunfield. MR**2328715**, DOI 10.1090/S0894-0347-07-00564-4 - S. Betley, J. H. Przytycki, and T. Żukowski,
*Hyperbolic structures on Dehn filling of some punctured-torus bundles over $S^1$*, Kobe J. Math.**3**(1987), no. 2, 117–147. MR**908780** - Chun Cao and G. Robert Meyerhoff,
*The orientable cusped hyperbolic $3$-manifolds of minimum volume*, Invent. Math.**146**(2001), no. 3, 451–478. MR**1869847**, DOI 10.1007/s002220100167 - Werner Fenchel,
*Elementary geometry in hyperbolic space*, De Gruyter Studies in Mathematics, vol. 11, Walter de Gruyter & Co., Berlin, 1989. With an editorial by Heinz Bauer. MR**1004006**, DOI 10.1515/9783110849455 - David Futer, Efstratia Kalfagianni, and Jessica S. Purcell,
*Dehn filling, volume, and the Jones polynomial*, J. Differential Geom.**78**(2008), no. 3, 429–464. MR**2396249** - David Gabai, G. Robert Meyerhoff, and Peter Milley,
*Volumes of tubes in hyperbolic 3-manifolds*, J. Differential Geom.**57**(2001), no. 1, 23–46. MR**1871490**
[GMM2]gmm2David Gabai, Robert Meyerhoff, and Peter Milley, Mom Technology and Volumes of Hyperbolic 3-manifolds, arXiv:math.GT/0606072.
- David Gabai, G. Robert Meyerhoff, and Nathaniel Thurston,
*Homotopy hyperbolic 3-manifolds are hyperbolic*, Ann. of Math. (2)**157**(2003), no. 2, 335–431. MR**1973051**, DOI 10.4007/annals.2003.157.335
[LM]LM M. Lackenby and R. Meyerhoff, The maximal number of exceptional Dehn surgeries, arXiv:0808.1176.
- Bruno Martelli and Carlo Petronio,
*Dehn filling of the “magic” 3-manifold*, Comm. Anal. Geom.**14**(2006), no. 5, 969–1026. MR**2287152**, DOI 10.4310/CAG.2006.v14.n5.a6 - S. V. Matveev and A. T. Fomenko,
*Isoenergetic surfaces of Hamiltonian systems, the enumeration of three-dimensional manifolds in order of growth of their complexity, and the calculation of the volumes of closed hyperbolic manifolds*, Uspekhi Mat. Nauk**43**(1988), no. 1(259), 5–22, 247 (Russian); English transl., Russian Math. Surveys**43**(1988), no. 1, 3–24. MR**937017**, DOI 10.1070/RM1988v043n01ABEH001554
[M]mmPeter Milley, Minimum volume hyperbolic 3-manifolds, - Harriet Moser,
*Proving a manifold to be hyperbolic once it has been approximated to be so*, Algebr. Geom. Topol.**9**(2009), no. 1, 103–133. MR**2471132**, DOI 10.2140/agt.2009.9.103 - Andrew Przeworski,
*A universal upper bound on density of tube packings in hyperbolic space*, J. Differential Geom.**72**(2006), no. 1, 113–127. MR**2215457** - William P. Thurston,
*Three-dimensional geometry and topology. Vol. 1*, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR**1435975**, DOI 10.1515/9781400865321
[W]wJeffrey Weeks,

*Journal of Topology*2009, doi: 10.1112/jtopol/jtp006. [M2]milPeter Milley. Source code to produce and check the volume bounds discussed in Section 4 are available free of charge by e-mail from Milley.

*SnapPea*, available from the author at www.geometrygames.org.

## Additional Information

**David Gabai**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 195365
**Robert Meyerhoff**- Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
**Peter Milley**- Affiliation: Department of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
- Received by editor(s): August 14, 2008
- Published electronically: May 1, 2009
- Additional Notes: The first author was partially supported by NSF grants DMS-0554374 and DMS-0504110.

THe second author was partially supported by NSF grants DMS-0553787 and DMS-0204311.

The third author was partially supported by NSF grant DMS-0554624 and by ARC Discovery grant DP0663399. - © Copyright 2009 by David Gabai, Robert Meyerhoff, and Peter Milley
- Journal: J. Amer. Math. Soc.
**22**(2009), 1157-1215 - MSC (2000): Primary 57M50; Secondary 51M10, 51M25
- DOI: https://doi.org/10.1090/S0894-0347-09-00639-0
- MathSciNet review: 2525782