Cusp shape and tunnel number

Dang, Vinh; Purcell, Jessica

doi:10.1090/proc/14336

Cusp shape and tunnel number
HTML articles powered by AMS MathViewer

by Vinh Dang and Jessica S. Purcell

Proc. Amer. Math. Soc. 147 (2019), 1351-1366

DOI: https://doi.org/10.1090/proc/14336

Published electronically: December 7, 2018

PDF | Request permission

Abstract:

We show that the set of cusp shapes of hyperbolic tunnel number one manifolds is dense in the Teichmüller space of the torus. A similar result holds for tunnel number $n$ manifolds. As a consequence, for fixed $n$, there are infinitely many hyperbolic tunnel number $n$ manifolds with at most one exceptional Dehn filling. This is in contrast to large volume Berge knots, which are tunnel number one manifolds, but with cusp shapes converging to a single point in Teichmüller space.

References

Colin C. Adams, Thrice-punctured spheres in hyperbolic $3$-manifolds, Trans. Amer. Math. Soc. 287 (1985), no. 2, 645–656. MR 768730, DOI 10.1090/S0002-9947-1985-0768730-6
Ian Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000), 431–449. MR 1799796, DOI 10.2140/gt.2000.4.431
Kenneth L. Baker, Surgery descriptions and volumes of Berge knots. II. Descriptions on the minimally twisted five chain link, J. Knot Theory Ramifications 17 (2008), no. 9, 1099–1120. MR 2457838, DOI 10.1142/S021821650800652X
Kenneth L. Baker, The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two, with an appendix by Neil R. Hoffman, arXiv:1504.06682, 2015.
Kenneth L. Baker, Cameron Gordon, and John Luecke, Bridge number and integral Dehn surgery, Algebr. Geom. Topol. 16 (2016), no. 1, 1–40. MR 3470696, DOI 10.2140/agt.2016.16.1
Steven A. Bleiler and Craig D. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), no. 3, 809–833. MR 1396779, DOI 10.1016/0040-9383(95)00040-2
Jeffrey F. Brock, Iteration of mapping classes and limits of hyperbolic 3-manifolds, Invent. Math. 143 (2001), no. 3, 523–570. MR 1817644, DOI 10.1007/PL00005799
Stephan D. Burton and Jessica S. Purcell, Geodesic systems of tunnels in hyperbolic 3-manifolds, Algebr. Geom. Topol. 14 (2014), no. 2, 925–952. MR 3160607, DOI 10.2140/agt.2014.14.925
R. D. Canary, D. B. A. Epstein, and P. L. Green, Notes on notes of Thurston [MR0903850], Fundamentals of hyperbolic geometry: selected expositions, London Math. Soc. Lecture Note Ser., vol. 328, Cambridge Univ. Press, Cambridge, 2006, pp. 1–115. With a new foreword by Canary. MR 2235710
Richard D. Canary, Marc Culler, Sa’ar Hersonsky, and Peter B. Shalen, Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups, J. Differential Geom. 64 (2003), no. 1, 57–109. MR 2015044
Daryl Cooper, Marc Lackenby, and Jessica S. Purcell, The length of unknotting tunnels, Algebr. Geom. Topol. 10 (2010), no. 2, 637–661. MR 2606795, DOI 10.2140/agt.2010.10.637
Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks, SnapPy, a computer program for studying the geometry and topology of $3$-manifolds, available at http://snappy.computop.org (2017).
Vinh Xuan Dang, Compression Bodies and Their Boundary Hyperbolic Structures, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–Brigham Young University. MR 3474635
Nathan M. Dunfield, Neil R. Hoffman, and Joan E. Licata, Asymmetric hyperbolic $L$-spaces, Heegaard genus, and Dehn filling, Math. Res. Lett. 22 (2015), no. 6, 1679–1698. MR 3507256, DOI 10.4310/MRL.2015.v22.n6.a7
Mario Eudave-Muñoz and J. Luecke, Knots with bounded cusp volume yet large tunnel number, J. Knot Theory Ramifications 8 (1999), no. 4, 437–446. MR 1697382, DOI 10.1142/S0218216599000304
David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Cusp areas of Farey manifolds and applications to knot theory, Int. Math. Res. Not. IMRN 23 (2010), 4434–4497. MR 2739802, DOI 10.1093/imrn/rnq037
John Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631–657. MR 1838999, DOI 10.1016/S0040-9383(00)00033-1
T. Jørgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), no. 1, 47–72. MR 1060898, DOI 10.7146/math.scand.a-12292
James Kaiser, Jessica S. Purcell, and Clint Rollins, Volumes of chain links, J. Knot Theory Ramifications 21 (2012), no. 11, 1250115, 17. MR 2969643, DOI 10.1142/S0218216512501155
Linda Keen, Bernard Maskit, and Caroline Series, Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets, J. Reine Angew. Math. 436 (1993), 209–219. MR 1207287
Linda Keen and Caroline Series, The Riley slice of Schottky space, Proc. London Math. Soc. (3) 69 (1994), no. 1, 72–90. MR 1272421, DOI 10.1112/plms/s3-69.1.72
Steven P. Kerckhoff and William P. Thurston, Noncontinuity of the action of the modular group at Bers’ boundary of Teichmüller space, Invent. Math. 100 (1990), no. 1, 25–47. MR 1037141, DOI 10.1007/BF01231179
Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243–282. MR 1756996, DOI 10.1007/s002220000047
Marc Lackenby and Jessica S. Purcell, Geodesics and compression bodies, Exp. Math. 23 (2014), no. 2, 218–240. MR 3223775, DOI 10.1080/10586458.2013.870503
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Joseph Maher and Saul Schleimer, The compression body graph has infinite diameter, arXiv:1803.06065, 2018.
Curt McMullen, Cusps are dense, Ann. of Math. (2) 133 (1991), no. 1, 217–247. MR 1087348, DOI 10.2307/2944328
John W. Morgan, On Thurston’s uniformization theorem for three-dimensional manifolds, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 37–125. MR 758464, DOI 10.1016/S0079-8169(08)61637-2
David Mumford, Caroline Series, and David Wright, Indra’s pearls, Cambridge University Press, New York, 2002. The vision of Felix Klein. MR 1913879, DOI 10.1017/CBO9781107050051.024
Hossein Namazi and Juan Souto, Non-realizability and ending laminations: proof of the density conjecture, Acta Math. 209 (2012), no. 2, 323–395. MR 3001608, DOI 10.1007/s11511-012-0088-0
Barbara E. Nimershiem, Isometry classes of flat $2$-tori appearing as cusps of hyperbolic $3$-manifolds are dense in the moduli space of the torus, Low-dimensional topology (Knoxville, TN, 1992) Conf. Proc. Lecture Notes Geom. Topology, III, Int. Press, Cambridge, MA, 1994, pp. 133–142. MR 1316178
Ken’ichi Ohshika, Realising end invariants by limits of minimally parabolic, geometrically finite groups, Geom. Topol. 15 (2011), no. 2, 827–890. MR 2821565, DOI 10.2140/gt.2011.15.827
Martin Scharlemann and Maggy Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006), 593–617. MR 2224466, DOI 10.2140/gt.2006.10.593
Masakazu Teragaito, A Seifert fibered manifold with infinitely many knot-surgery descriptions, Int. Math. Res. Not. IMRN 9 (2007), Art. ID rnm 028, 16. MR 2347296, DOI 10.1093/imrn/rnm028
W. Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Notes, 1979.