The lowest volume 3–orbifolds with high torsion

Atkinson, Christopher; Futer, David

doi:10.1090/tran/6920

The lowest volume $3$–orbifolds with high torsion
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by Christopher K. Atkinson and David Futer PDF

Trans. Amer. Math. Soc. 369 (2017), 5809-5827

Abstract:

For each natural number $n \geq 4$, we determine the unique lowest volume hyperbolic $3$–orbifold whose torsion orders are bounded below by $n$. This lowest volume orbifold has base space the $3$–sphere and singular locus the figure–$8$ knot, marked $n$. We apply this result to give sharp lower bounds on the volume of a hyperbolic manifold in terms of the order of elements in its symmetry group.

References

John W. Aaber and Nathan Dunfield, Closed surface bundles of least volume, Algebr. Geom. Topol. 10 (2010), no. 4, 2315–2342. MR 2745673, DOI 10.2140/agt.2010.10.2315
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
Colin C. Adams, Limit volumes of hyperbolic three-orbifolds, J. Differential Geom. 34 (1991), no. 1, 115–141. MR 1114455
Colin C. Adams, Noncompact hyperbolic $3$-orbifolds of small volume, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 1–15. MR 1184398
Ian Agol, The minimal volume orientable hyperbolic 2-cusped 3-manifolds, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3723–3732. MR 2661571, DOI 10.1090/S0002-9939-10-10364-5
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, 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
Roger C. Alperin, An elementary account of Selberg’s lemma, Enseign. Math. (2) 33 (1987), no. 3-4, 269–273. MR 925989
Christopher K. Atkinson and David Futer, Ancillary files stored with the arXiv version of this paper, http://arxiv.org/src/1507.07894/anc.
Christopher K. Atkinson and David Futer, Small volume link orbifolds, Math. Res. Lett. 20 (2013), no. 6, 995–1016. MR 3228616, DOI 10.4310/MRL.2013.v20.n6.a1
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
Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, vol. 5, Mathematical Society of Japan, Tokyo, 2000. With a postface by Sadayoshi Kojima. MR 1778789
William D. Dunbar and G. Robert Meyerhoff, Volumes of hyperbolic $3$-orbifolds, Indiana Univ. Math. J. 43 (1994), no. 2, 611–637. MR 1291531, DOI 10.1512/iumj.1994.43.43025
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 Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Symmetric links and Conway sums: volume and Jones polynomial, Math. Res. Lett. 16 (2009), no. 2, 233–253. MR 2496741, DOI 10.4310/MRL.2009.v16.n2.a3
David Gabai, Robert Meyerhoff, and Peter Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009), no. 4, 1157–1215. MR 2525782, DOI 10.1090/S0894-0347-09-00639-0
F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3611–3643. MR 1433117, DOI 10.1090/S0002-9947-97-01989-2
F. W. Gehring, T. H. Marshall, and G. J. Martin, The spectrum of elliptic axial distances in Kleinian groups, Indiana Univ. Math. J. 47 (1998), no. 1, 1–10. MR 1631604, DOI 10.1512/iumj.1998.47.1433
F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups, J. Anal. Math. 63 (1994), 175–219. MR 1269219, DOI 10.1007/BF03008423
Frederick W. Gehring and Gaven J. Martin, Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group, Ann. of Math. (2) 170 (2009), no. 1, 123–161. MR 2521113, DOI 10.4007/annals.2009.170.123
Oliver Goodman, Snap, a computer program for studying arithmetic invariants of hyperbolic 3-manifolds. Includes Tube, a program for computing tube domains, http://www.ms.unimelb.edu.au/˜snap/.
Craig D. Hodgson and Steven P. Kerckhoff, The shape of hyperbolic Dehn surgery space, Geom. Topol. 12 (2008), no. 2, 1033–1090. MR 2403805, DOI 10.2140/gt.2008.12.1033
A. Hurwitz, Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1892), no. 3, 403–442 (German). MR 1510753, DOI 10.1007/BF01443420
A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90–96 (1961). MR 146724
T. H. Marshall and G. J. Martin, Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group, Ann. of Math. (2) 176 (2012), no. 1, 261–301. MR 2925384, DOI 10.4007/annals.2012.176.1.4
Robert Meyerhoff, Sphere-packing and volume in hyperbolic $3$-space, Comment. Math. Helv. 61 (1986), no. 2, 271–278. MR 856090, DOI 10.1007/BF02621915
Peter Milley, Minimum volume hyperbolic 3-manifolds, J. Topol. 2 (2009), no. 1, 181–192. MR 2499442, DOI 10.1112/jtopol/jtp006
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
Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI 10.1016/0040-9383(85)90004-7
William P. Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Notes, 1980.
A. Yu. Vesnin and A. D. Mednykh, Hyperbolic volumes of Fibonacci manifolds, Sibirsk. Mat. Zh. 36 (1995), no. 2, 266–277, i (Russian, with Russian summary); English transl., Siberian Math. J. 36 (1995), no. 2, 235–245. MR 1340395, DOI 10.1007/BF02110146