Effective distance between nested Margulis tubes

Futer, David; Purcell, Jessica; Schleimer, Saul

doi:10.1090/tran/7678

Effective distance between nested Margulis tubes
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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/.