Effective distance between nested Margulis tubes
HTML articles powered by AMS MathViewer
- by David Futer, Jessica S. Purcell and Saul Schleimer PDF
- Trans. Amer. Math. Soc. 372 (2019), 4211-4237
Abstract:
We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic $3$–manifold.References
- Tarik Aougab, Priyam Patel, and Samuel J. Taylor, Covers of surfaces, Kleinian groups, and the curve complex, arXiv:1810.12953, 2018.
- Tarik Aougab, Samuel J. Taylor, and Richard C.H. Webb, Effective Masur–Minsky distance formulas and applications to hyperbolic $3$–manifolds, https://math.temple.edu/~samuel.taylor/Main.pdf.
- Brian H. Bowditch, The ending lamination theorem, http://homepages.warwick.ac.uk/~masgak/ papers/elt.pdf.
- Jeffrey F. Brock, Weil-Petersson translation distance and volumes of mapping tori, Comm. Anal. Geom. 11 (2003), no. 5, 987–999. MR 2032506, DOI 10.4310/CAG.2003.v11.n5.a6
- Jeffrey F. Brock, The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003), no. 3, 495–535. MR 1969203, DOI 10.1090/S0894-0347-03-00424-7
- Jeffrey F. Brock and Kenneth W. Bromberg, On the density of geometrically finite Kleinian groups, Acta Math. 192 (2004), no. 1, 33–93. MR 2079598, DOI 10.1007/BF02441085
- Jeffrey F. Brock and Kenneth W. Bromberg, Inflexibility, Weil-Peterson distance, and volumes of fibered 3-manifolds, Math. Res. Lett. 23 (2016), no. 3, 649–674. MR 3533189, DOI 10.4310/MRL.2016.v23.n3.a4
- Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2) 176 (2012), no. 1, 1–149. MR 2925381, DOI 10.4007/annals.2012.176.1.1
- Robert Brooks and J. Peter Matelski, Collars in Kleinian groups, Duke Math. J. 49 (1982), no. 1, 163–182. MR 650375
- C. Cao, F. W. Gehring, and G. J. Martin, Lattice constants and a lemma of Zagier, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 107–120. MR 1476983, DOI 10.1090/conm/211/02816
- Daryl Cooper, David Futer, and Jessica S. Purcell, Dehn filling and the geometry of unknotting tunnels, Geom. Topol. 17 (2013), no. 3, 1815–1876. MR 3073937, DOI 10.2140/gt.2013.17.1815
- Marc Culler and Peter B. Shalen, The volume of a hyperbolic $3$-manifold with Betti number $2$, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1281–1288. MR 1205485, DOI 10.1090/S0002-9939-1994-1205485-6
- Marc Culler and Peter B. Shalen, Margulis numbers for Haken manifolds, Israel J. Math. 190 (2012), 445–475. MR 2956250, DOI 10.1007/s11856-011-0189-z
- David Futer, Jessica S. Purcell, and Saul Schleimer, Effective bilipschitz bounds on drilling and filling, in preparation.
- David Futer and Saul Schleimer, Cusp geometry of fibered 3-manifolds, Amer. J. Math. 136 (2014), no. 2, 309–356. MR 3188063, DOI 10.1353/ajm.2014.0012
- 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
- Sebastian Hensel, Piotr Przytycki, and Richard C. H. Webb, 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 755–762. MR 3336835, DOI 10.4171/JEMS/517
- Craig D. Hodgson and Steven P. Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Ann. of Math. (2) 162 (2005), no. 1, 367–421. MR 2178964, DOI 10.4007/annals.2005.162.367
- 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
- Robert Meyerhoff, A lower bound for the volume of hyperbolic $3$-manifolds, Canad. J. Math. 39 (1987), no. 5, 1038–1056. MR 918586, DOI 10.4153/CJM-1987-053-6
- Yair Minsky, The classification of Kleinian surface groups. I. Models and bounds, Ann. of Math. (2) 171 (2010), no. 1, 1–107. MR 2630036, DOI 10.4007/annals.2010.171.1
- Yair N. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), no. 2, 559–626. MR 1689341, DOI 10.2307/120976
- 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
- Ken’ichi Ohshika, Kleinian groups which are limits of geometrically finite groups, Mem. Amer. Math. Soc. 177 (2005), no. 834, xii+116. MR 2154090, DOI 10.1090/memo/0834
- Peter B. Shalen, A generic Margulis number for hyperbolic 3-manifolds, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 103–109. MR 2866926, DOI 10.1090/conm/560/11094
- Peter B. Shalen, Small optimal Margulis numbers force upper volume bounds, Trans. Amer. Math. Soc. 365 (2013), no. 2, 973–999. MR 2995380, DOI 10.1090/S0002-9947-2012-05657-1
- Perry Susskind, The Margulis region and continued fractions, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) Contemp. Math., vol. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 335–343. MR 1940179, DOI 10.1090/conm/311/05462
- William P. Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Notes, 1980, http://www.msri.org/gt3m/.
Additional Information
- David Futer
- Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
- MR Author ID: 671567
- ORCID: 0000-0002-2595-6274
- Email: dfuter@temple.edu
- Jessica S. Purcell
- Affiliation: School of Mathematical Sciences, Monash University, Victoria 3800, Australia
- MR Author ID: 807518
- ORCID: 0000-0002-0618-2840
- Email: jessica.purcell@monash.edu
- Saul Schleimer
- Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 689853
- Email: s.schleimer@warwick.ac.uk
- Received by editor(s): January 22, 2018
- Received by editor(s) in revised form: August 14, 2018
- Published electronically: May 30, 2019
- Additional Notes: The first author was supported in part by NSF grant DMS–1408682
The second author was supported in part by the Australian Research Council
All three authors acknowledge support from NSF grants DMS–1107452, 1107263, 1107367, “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network), which funded an international trip to collaborate on this paper
This work is in the public domain. - Journal: Trans. Amer. Math. Soc. 372 (2019), 4211-4237
- MSC (2010): Primary 57M50, 30F40
- DOI: https://doi.org/10.1090/tran/7678
- MathSciNet review: 4009389