Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation


Authors: R. Roberts, J. Shareshian and M. Stein
Journal: J. Amer. Math. Soc. 16 (2003), 639-679
MSC (2000): Primary 57M25; Secondary 57R30
DOI: https://doi.org/10.1090/S0894-0347-03-00426-0
Published electronically: March 3, 2003
MathSciNet review: 1969207
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented $\mathbb R$-order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.


References [Enhancements On Off] (What's this?)

  • [Ag00] I. Agol, Bounds on exceptional Dehn filling, Geom. Top. 4 (2000), 431-449. MR 2001j:57019
  • [Ba89] M. Baker, Covers of Dehn fillings on once-punctured torus bundles, Proc. A.M.S. 105(3) (1989), 747-754. MR 90e:57001
  • [Ba90] M. Baker, Covers of Dehn fillings on once-punctured torus bundles,II, Proc. A.M.S. 110(4) (1990), 1099-1108. MR 91e:57004
  • [Ba98] T. Barbot, Actions de groupes sur les 1-variétés non séparées et feuilletages de codimension un (French)[Actions of groups on non-Hausdorff 1-manifolds and codimension one foliations] Ann. Fac. Sci. Toulouse Math. (6)7 (1998), no.4, 559-597. MR 2000e:57045
  • [Be99] S. Berberian, Fundamentals of real analysis, New York : Springer, 1999. MR 99i:28001
  • [BPZ] S. Betley, J. Przytycki, and T. Zukowski, Hyperbolic structures on dehn filling of some punctured-torus bundles over $S^1$, Kobe J. Math. 3 (1986), 117-147. MR 89f:57017
  • [BH96] S. Bleiler and C. Hodgson, Spherical space forms and dehn filling, Topology 35(3) (1996), 809-833. MR 97f:57007
  • [BMR] B. Bowditch, C. Maclachlan, and A. Reid, Arithmetic hyperbolic surface bundles, Math. Ann. 302 (1995), 31-60. MR 96a:57032
  • [Br93] M. Brittenham, Essential laminations in Seifert-fibered spaces, Topology 32 (1993), 61-85. MR 94c:57027
  • [BNR] M. Brittenham, R. Naimi and R. Roberts, Graph manifolds and taut foliations, J. Diff. Geom. 45 (1997), 446-470. MR 98j:57040
  • [Ca] D. Calegari, Promoting essential laminations, I, preprint.
  • [CD03] D. Calegari and N. Dunfield, Laminations and groups of homeomorphisms of the circle, Inv. Math., to appear.
  • [CC] A. Candel and L. Conlon, Foliations II, preprint.
  • [Ch01] I. Chiswell, Introduction to $\Lambda$-trees, World Scientific, Singapore, 2001.
  • [Cl91] E. Claus, Essential laminations in closed Seifert-fibered spaces, Thesis, University of Texas at Austin, 1991.
  • [CHK] D. Cooper, C. Hodgson, and S. Kerckhoff, Three-dimensional orbifolds and cone manifolds, MSJ Memoirs 5 (2000). MR 2002c:57027
  • [CJR] M. Culler, W. Jaco and H. Rubinstein, Incompressible surfaces in once-punctured torus bundles, Proc. L.M.S. 45 (1982), 385-419. MR 84a:57010
  • [CM87] M. Culler and J. Morgan, Group actions on $\mathbb R$-trees, Proc. L.M.S. (3) 55 (1987), 571-604. MR 88f:20055
  • [CV96] M. Culler and K. Vogtmann, A group theoretic criterion for property FA, Proc. A.M.S. 124(3) (1996), 676-683. MR 96f:20040
  • [De32] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl. 11(1932), 333-375.
  • [Fe] S. Fenley, Pseudo-Anosov flows and incompressible tori, preprint.
  • [FH82] W. Floyd and A. Hatcher, Incompressible surfaces in punctured-torus bundles, Topology Appl. 13(3) (1982), 263-282. MR 83h:57015
  • [Ga92] D. Gabai, Taut foliations of 3-manifolds and suspensions of $S^1$, Ann. Inst. Fourier, Grenoble, 42(1-2) (1992), 193-208. MR 93d:57028
  • [Ga97] D. Gabai, Problems in foliations and laminations, Studies in Advanced Math. 2(2) (1997), 1-33.
  • [Ga98] D. Gabai, Quasi-minimal semi-Euclidean laminations in 3-manifolds, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), 195-242, Int. Press, Boston, MA, 1998. MR 2000b:57022
  • [GK97] D. Gabai and W. Kazez, Order trees and essential laminations of the plane, Mathematical Research Letters 4 (1997), 603-616. MR 98k:57024
  • [GK98] D. Gabai and W. Kazez, Group negative curvature for 3-manifolds with genuine laminations, Geom. Topol. 2 (1998), 65-77. MR 99e:57023
  • [GO89] D. Gabai and U. Oertel, Essential laminations in 3-manifolds, Ann. Math. 130 (1989), 41-73. MR 90h:57012
  • [Go75] S. Goodman, Closed leaves in foliations of codimension one, Comm. Math. Helv. 50 (1975), 383-388. MR 54:11350
  • [HR57] A. Haefliger and G. Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, Ens. Math. 3 (1957), 107-125. MR 19:671c
  • [Ha92] A. Hatcher, Some examples of essential laminations in 3-manifolds, Ann. Inst. Fourier (Grenoble) 42(1-2) (1992), 313-325. MR 93e:57026
  • [HMW] C. Hodgson, R. Meyerhoff, and J. Weeks, Surgeries on the Whitehead link yield geometrically similar matrices, Topology '90, 1992, 195-206. MR 93i:57019
  • [Jo77] T. Jorgensen, Compact 3-manifolds of constant negative curvature, Ann. Math. 106 (1977), 61-72. MR 56:8840
  • [La00] M. Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140(2) (2000), 243-282. MR 2001m:57003
  • [Ma00] J. Masters, Virtual homology of surgered torus bundles, Pac. J. Math. 195(1) (2000), 205-223. MR 2001g:57037
  • [Mc87] D. McCullough, Automorphisms of punctured-surface bundles, Lecture Notes in Pure and Appl. Math. 105 (1987), 179-209. MR 88c:57014
  • [Mo92] J. Morgan, $\Lambda$-trees and their applications, Bull. A.M.S. 26(1) (1992), 87-112. MR 92e:20017
  • [MS88] J. Morgan and P. Shalen, Degenerations of hyperbolic structures, III: actions of 3-manifold groups on trees and Thurston's compactification theorem, Ann. Math. 127 (1988), 457-519. MR 89e:57010b
  • [Ni17] J. Nielsen, Die Isomorphismengruppe der allgemeinen unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1917), 385-397.
  • [No65] S. Novikov, `Topology of foliations', Trans. Moscow Math. Soc. 14 (1965), 248-278. MR 34:824
  • [NR92] W. Neumann and A. Reid, Arithmetic of hyperbolic 3-manifolds, Topology '90, 1992, 273-310. MR 94c:57024
  • [Pa78] C. Palmeira, Open manifolds foliated by planes, Ann. Math. 107 (1978), 109-131. MR 58:18490
  • [Pa95] F. Paulin, Actions de groupes sur les arbres, Sém. Bourbaki, 1995-1996, no. 808. MR 98j:20028
  • [Pr83] J. Przytycki, Nonorientable, incompressible surfaces of genus 3 in $M_{\varphi}({\frac{\lambda}{\mu}})$ manifolds, Collect. Math. 34(1) (1983), 37-79. MR 86e:57015
  • [Pr84] J. Przytycki, Incompressibility of surfaces with four boundary components after Dehn surgery, Demonstratio Math. 17(1) (1984), 119-126. MR 86a:57018
  • [Re95] A. Reid, A non-Haken hyperbolic 3-manifold covered by a surface bundle, Pac. J. Math. 167(1) (1995), 163-182. MR 95m:57025
  • [RW99] A. Reid and S. Wang, Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree, Comm. Anal. Geom. 7(1) (1999), 105-132. MR 2000c:57042
  • [RSS] R. Roberts, J. Shareshian and M. Stein, in preparation.
  • [RS01] R. Roberts and M. Stein, Group actions on order trees, Top. Appl. 115 (2001), 175-201. MR 2002f:57001
  • [Ro77] D. Rolfsen, Knots and Links, Publish or Perish, 1977. MR 58:24236
  • [Sc74] P. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Annals of Math. 100(2) (1974), 386-400. MR 50:8557
  • [Sh87] P. Shalen, Dendrology of groups: An introduction, in Essays in group theory, M.S.R.I. Publ. 8, Springer, Berlin, 1987. MR 89d:57012
  • [Sh91] P. Shalen, Dendrology and its applications, in Group theory from a geometrical viewpoint, World Scientific, Singapore, 1991. MR 94e:57020
  • [Se77] J.-P. Serre, Trees, Springer Verlag, 1980, Translation by J. Stillwell of Arbres, Amalgams, $SL_2$, Astérisque 46, Soc. Math. France (1977). MR 82c:20083
  • [Th79] W. Thurston, The geometry and topology of three-manifolds, Princeton, 1979.
  • [Th82] W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. A.M.S. 6(3) (1982), 357-381. MR 83h:57019
  • [Th88] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S. 19(2) (1988), 417-431. MR 89k:57023
  • [We85] J. Weeks, Hyperbolic structures on three-manifolds, Thesis, Princeton University, 1985.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 57M25, 57R30

Retrieve articles in all journals with MSC (2000): 57M25, 57R30


Additional Information

R. Roberts
Affiliation: Department of Mathematics, Washington University, St Louis, Missouri 63130
Email: roberts@math.wustl.edu

J. Shareshian
Affiliation: Department of Mathematics, Washington University, St Louis, Missouri 63130
Email: shareshi@math.wustl.edu

M. Stein
Affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106
Email: Melanie.Stein@mail.cc.trincoll.edu

DOI: https://doi.org/10.1090/S0894-0347-03-00426-0
Keywords: Hyperbolic, 3-manifold, tree, order tree, 1-manifold, group action, punctured surface bundle, Reebless foliation, essential lamination
Received by editor(s): June 2, 2002
Published electronically: March 3, 2003
Additional Notes: The first author was partially supported by National Science Foundation grant DMS-9971333
The second author was partially supported by National Science Foundation grant DMS-0070757
The third author was partially supported by a Trinity College Faculty Research Grant
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society