Small optimal Margulis numbers force upper volume bounds
HTML articles powered by AMS MathViewer
- by Peter B. Shalen PDF
- Trans. Amer. Math. Soc. 365 (2013), 973-999 Request permission
Abstract:
If $\lambda$ is a positive real number strictly less than $\operatorname {log}3$, there is a positive number $V_\lambda$ such that every orientable hyperbolic $3$-manifold of volume greater than $V_\lambda$ admits $\lambda$ as a Margulis number. If $\lambda <(\operatorname {log}3)/2$, such a $V_\lambda$ can be specified explicitly and is bounded above by \[ \lambda \bigg (6+\frac {880}{\operatorname {log}3-2\lambda } \operatorname {log}{1\over \operatorname {log}3-2\lambda }\bigg ),\] where $\operatorname {log}$ denotes the natural logarithm. These results imply that for $\lambda <\operatorname {log}3$, an orientable hyperbolic $3$-manifold that does not have $\lambda$ as a Margulis number has a rank-$2$ subgroup of bounded index in its fundamental group and in particular has a fundamental group of bounded rank. Again, the bounds in these corollaries can be made explicit if $\lambda <(\operatorname {log}3)/2$.References
- Ian Agol. Tameness of hyperbolic 3-manifolds. arXiv:math.GT/0405568.
- 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 and Yi Liu. Presentation length and Simon’s conjecture. arXiv:1006.5262.
- James W. Anderson, Richard D. Canary, Marc Culler, and Peter B. Shalen, Free Kleinian groups and volumes of hyperbolic $3$-manifolds, J. Differential Geom. 43 (1996), no. 4, 738–782. MR 1412683
- Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992. MR 1219310, DOI 10.1007/978-3-642-58158-8
- Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385–446. MR 2188131, DOI 10.1090/S0894-0347-05-00513-8
- 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
- Daryl Cooper, The volume of a closed hyperbolic $3$-manifold is bounded by $\pi$ times the length of any presentation of its fundamental group, Proc. Amer. Math. Soc. 127 (1999), no. 3, 941–942. MR 1646313, DOI 10.1090/S0002-9939-99-05190-4
- Marc Culler and Peter B. Shalen. Margulis numbers for Haken manifolds. Accepted Israel Journal of Mathematics.
- Marc Culler and Peter B. Shalen, Paradoxical decompositions, $2$-generator Kleinian groups, and volumes of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 231–288. MR 1135928, DOI 10.1090/S0894-0347-1992-1135928-4
- D. B. A. Epstein, Projective planes in $3$-manifolds, Proc. London Math. Soc. (3) 11 (1961), 469–484. MR 152997, DOI 10.1112/plms/s3-11.1.469
- R. Mark Goresky, Triangulation of stratified objects, Proc. Amer. Math. Soc. 72 (1978), no. 1, 193–200. MR 500991, DOI 10.1090/S0002-9939-1978-0500991-2
- Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
- Jack E. Graver, An analytic triangulation of an arbitrary real analytic variety, J. Math. Mech. 13 (1964), 1021–1036. MR 0167989
- John Hempel, 3-manifolds, AMS Chelsea Publishing, Providence, RI, 2004. Reprint of the 1976 original. MR 2098385, DOI 10.1090/chel/349
- Morris W. Hirsch, Smooth regular neighborhoods, Ann. of Math. (2) 76 (1962), 524–530. MR 149492, DOI 10.2307/1970372
- William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
- Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
- 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
- Troels Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), no. 3, 739–749. MR 427627, DOI 10.2307/2373814
- Troels Jørgensen and Peter Klein, Algebraic convergence of finitely generated Kleinian groups, Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 131, 325–332. MR 668178, DOI 10.1093/qmath/33.3.325
- Wilhelm Magnus, Über freie Faktorgruppen und freie Untergruppen gegebener Gruppen, Monatsh. Math. Phys. 47 (1939), no. 1, 307–313 (German). MR 1550819, DOI 10.1007/BF01695503
- William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
- 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
- Peter Milley, Minimum volume hyperbolic 3-manifolds, J. Topol. 2 (2009), no. 1, 181–192. MR 2499442, DOI 10.1112/jtopol/jtp006
- Edwin E. Moise, Affine structures in $3$-manifolds. II. Positional properties of $2$-spheres, Ann. of Math. (2) 55 (1952), 172–176. MR 45380, DOI 10.2307/1969426
- James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2) 72 (1960), 521–554. MR 121804, DOI 10.2307/1970228
- H. Namazi and J. Souto. Nonrealizability in handlebodies and ending laminations. preprint.
- Ken’ichi Ohshika. Realizing end invariants by limits of minimally parabolic, geometrically finite groups. arXiv:math.DG/0504506.
- G. P. Scott, Compact submanifolds of $3$-manifolds, J. London Math. Soc. (2) 7 (1973), 246–250. MR 326737, DOI 10.1112/jlms/s2-7.2.246
- Caroline Series, A crash course on Kleinian groups, Rend. Istit. Mat. Univ. Trieste 37 (2005), no. 1-2, 1–38 (2006). MR 2227047
- Peter B. Shalen. A generic Margulis number for hyperbolic 3-manifolds. arXiv:1008.4081.
- Peter B. Shalen. Trace fields and Margulis numbers. In preparation.
- Peter B. Shalen and Philip Wagreich, Growth rates, $Z_p$-homology, and volumes of hyperbolic $3$-manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 2, 895–917. MR 1156298, DOI 10.1090/S0002-9947-1992-1156298-8
- Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
- Jeffrey Renwick Weeks, HYPERBOLIC STRUCTURES ON THREE-MANIFOLDS (DEHN SURGERY, KNOT, VOLUME), ProQuest LLC, Ann Arbor, MI, 1985. Thesis (Ph.D.)–Princeton University. MR 2634492
- J. H. C. Whitehead, Manifolds with transverse fields in euclidean space, Ann. of Math. (2) 73 (1961), 154–212. MR 124917, DOI 10.2307/1970286
Additional Information
- Peter B. Shalen
- Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045
- MR Author ID: 159535
- Email: shalen@math.uic.edu
- Received by editor(s): October 13, 2010
- Received by editor(s) in revised form: June 16, 2011
- Published electronically: July 25, 2012
- Additional Notes: This work was partially supported by NSF grant DMS-0906155
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 973-999
- MSC (2010): Primary 57M50
- DOI: https://doi.org/10.1090/S0002-9947-2012-05657-1
- MathSciNet review: 2995380
Dedicated: Dedicated to José Montesinos on the occasion of his 65th birthday