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)



Foliations with good geometry

Author: Sérgio R. Fenley
Journal: J. Amer. Math. Soc. 12 (1999), 619-676
MSC (1991): Primary 53C12, 53C23, 57R30, 58F15, 58F18; Secondary 53C22, 57M99, 58F25
Published electronically: April 26, 1999
MathSciNet review: 1674739
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The goal of this article is to show that there is a large class of closed hyperbolic 3-manifolds admitting codimension one foliations with good large scale geometric properties. We obtain results in two directions. First there is an internal result: A possibly singular foliation in a manifold is quasi-isometric if, when lifted to the universal cover, distance along leaves is efficient up to a bounded multiplicative distortion in measuring distance in the universal cover. This means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight-this is the best geometric behavior. We previously proved that nonsingular codimension one foliations in closed hyperbolic 3-manifolds can never be quasi-isometric. In this article we produce a large class of singular quasi-isometric, codimension one foliations in closed hyperbolic 3-manifolds. The foliations are stable and unstable foliations of pseudo-Anosov flows. Our second result is an external result, relating (nonsingular) foliations in hyperbolic 3-manifolds with their limit sets in the universal cover, that is, showing that leaves in the universal cover have good asymptotic behavior. Let $\mathcal G$ be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then $\mathcal G$ is not quasi-isometric, but suppose that $\mathcal G$ is transverse to a quasigeodesic pseudo-Anosov flow with quasi-isometric stable and unstable foliations-which are given by the internal result. We then show that the lifts of leaves of $\mathcal G$ to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis. The main tools used to prove these results are first a link between geometric properties of stable/unstable foliations of pseudo-Anosov flows and the topology of these foliations in the universal cover, and second a topological theory of the joint structure of the pseudo-Anosov foliation in the universal cover.

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

  • [An] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1969). MR 39:3527
  • [An-Si] D. V. Anosov and Y. Sinai, Some smoothly ergodic systems, Russian Math. Surveys 22 (1967) 5 103-167. MR 37:370
  • [As] D. Asimov, Round handles and non-singular Morse-Smale flows, Ann. of Math. 102 (1979) 41-54. MR 52:1780
  • [Ba] T. Barbot, Flots d'Anosov sur les variétés graphées au sens de Waldhausen", Ann. Inst. Fourier Grenoble 46 (1996) 1451-1517. MR 97j:57031
  • [Be-Me] M. Bestvina and G. Mess, The boundary of negatively curved groups, Jour. Amer. Math. Soc., 4 (1991) 469-481. MR 93j:20076
  • [Bl-Ca] S. Bleiler and A. Casson, Automorphisms of surfaces after Nielsen and Thurston, Cambridge Univ. Press, 1988. MR 89k:57025
  • [Bo] F. Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986) 71-158. MR 88c:57013
  • [Br] M. Brittenham, Essential laminations in Seifert fibered spaces, Topology 32 (1993) 61-85. MR 94c:57027
  • [BNR] M. Brittenham, R. Naimi, R. Roberts, Graph manifolds and taut foliations, Jour. Diff. Geom. 45 (1997) 446-470. MR 98j:57040
  • [Can] A. Candel, Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. (4) 26 (1993) 489-516. MR 94f:57025
  • [Ca-Th] J. Cannon and W. Thurston, Group invariant Peano curves, to appear.
  • [Ca-Co] J. Cantwell and L. Conlon, Smoothability of proper foliations, Ann. Inst. Fourier, Grenoble, 38 (1988) 403-453. MR 90f:57034
  • [CLR1] D. Cooper, D. Long and A. Reid, Bundles and finite foliations, Inven. Math. 118 (1994) 255-283. MR 96h:57013
  • [CLR2] D. Cooper, D. Long and A. Reid, Finite foliations and similarity interval exchange maps, Topology, 36 (1997) 209-227. MR 97j:57032
  • [CDP] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Les groupes hyperboliques de Gromov, Lecture Notes in Math., 1441 Springer Verlag (1991).
  • [Fe1] S. Fenley, Asymptotic properties of depth one foliations in hyperbolic $3$-manifolds, Jour. Diff. Geom., 36 (1992) 269-313. MR 93k:57030
  • [Fe2] S. Fenley, Quasi-isometric foliations, Topology 31 (1992) 667-676. MR 94a:57044
  • [Fe3] S. Fenley, Anosov flows in $3$-manifolds, Ann. of Math. 139 (1994) 79-115. MR 94m:58162
  • [Fe4] S. Fenley, Quasigeodesic Anosov flows and homotopic properties of closed orbits, Jour. Diff. Geo. 41 (1995) 479-514. MR 96f:58118
  • [Fe5] S. Fenley, One sided branching in Anosov foliations, Comm. Math. Helv. 70 (1995) 248-266. MR 96c:57052
  • [Fe6] S. Fenley, The structure of branching in Anosov flows of $3$-manifolds, Comm. Math. Helv. 73 (1998) 259-297. MR 99a:58123
  • [Fe7] S. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in $3$-manifolds, to appear in Topology (1999).
  • [Fe-Mo] S. Fenley and L. Mosher, Quasigeodesic flows in hyperbolic $3$-manifolds, to appear in Topology.
  • [Fr-Wi] J. Franks and R. Williams, Anomalous Anosov flows, in Global theory of Dyn. Systems, Lecture Notes in Math. 819 Springer (1980). MR 82e:58078
  • [Ga1] D. Gabai, Foliations and the topology of 3-manifolds, J. Diff. Geo. 18 (1983) 445-503. MR 86a:57009
  • [Ga2] D. Gabai, Foliations and the topology of 3-manifolds II, J. Diff. Geo. 26 (1987) 461-478. MR 89a:57014a
  • [Ga3] D. Gabai, Foliations and the topology of 3-manifolds III, J. Diff. Geo. 26 (1987) 479-536. MR 89a:57014b
  • [Ga-Oe] D. Gabai and U. Oertel, Essential laminations and $3$-manifolds, Ann. of Math. 130 (1989) 41-73. MR 90h:57012
  • [Gh-Ha] E. Ghys and P. de la Harpe, eds., Sur les groupes hyperboliques d'aprés Mikhael Gromov, Progress in Math., 83 Birkhäuser, 1991. MR 92f:53050
  • [Gr] M. Gromov, Hyperbolic groups, in Essays on group theory, 75-263, Springer, 1987. MR 89e:20070
  • [He] J. Hempel, 3-manifolds, Ann. of Math. Studies 86, Princeton University Press, 1976. MR 54:3702
  • [Ja-Sh] W. Jaco and P. Shalen, Seifert fibered spaces in $3$-manifolds, Memoirs of the A. M. S. 21, number 220, 1979. MR 81c:57010
  • [Le] G. Levitt, Foliations and laminations on hypebolic surfaces, Topology 22 (1983) 119-135. MR 84h:57015
  • [Ma] A. Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. 99 (1974) 383-462. MR 50:2485
  • [Mor] J. Morgan, On Thurston's uniformization theorem for $3$-dimensional manifolds, in The Smith Conjecture, J. Morgan and H. Bass, eds., Academic Press, New York, 1984, 37-125. CMP 17:01
  • [Mo1] L. Mosher, Dynamical systems and the homology norm of a $3$-manifold I. Efficient interesection of surfaces and flows, Duke Math. Jour. 65 (1992) 449-500. MR 93g:57018a
  • [Mo2] L. Mosher, Dynamical systems and the homology norm of a $3$-manifold II, Invent. Math. 107 (1992) 243-281. MR 93g:57018b
  • [Mo3] L. Mosher, Examples of quasigeodesic flows on hyperbolic $3$-manifolds, in Proceedings of the Ohio State University Research Semester on Low-Dimensional topology, W. de Gruyter, 1992. MR 93i:58120
  • [Mo4] L. Mosher, Laminations and flows transverse to finite depth foliations, manuscript available from mosher/, Part I: Branched surfaces and dynamics, Part II in preparation.
  • [No] S. P. Novikov, Topology of foliations, Trans. Moscow Math. Soc. 14 (1963) 268-305. MR 34:824
  • [Pa] C. Palmeira, Open manifolds foliated by planes, Ann. of Math., 107 (1978) 109-131. MR 58:18490
  • [Pl] J. Plante, Foliations with measure preserving holonomy, Ann. of Math. 107 (1975) 327-361. MR 52:11947
  • [Ro] H. Rosenberg, Foliations by planes, Topology 7 (1968) 131-138. MR 37:3595
  • [St] K. Strebel, Quadratic differentials, Springer Verlag, 1984. MR 86a:30072
  • [Sc] P. Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983) 401-487. MR 84m:57009
  • [Su] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Inven. Math. 36 (1976) 225-255. MR 55:6440
  • [Th1] W. Thurston, Foliations of $3$-manifolds which are Circle Bundles, Thesis, University of California, Berkeley, 1972.
  • [Th2] W. Thurston, The geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1982.
  • [Th3] W. Thurston, Hyperbolic structures on $3$-manifolds II: Surface groups and $3$-manifolds that fiber over the circle, preprint.
  • [Th4] W. Thurston, 3-dimensional manifolds, Kleinian Groups and hyperbolic geometry, Bull. A.M.S., 6 (new series), (1982) 357-381. MR 83h:57019
  • [Ze] A. Zeghib, Sur les feuilletages géodésiques continus des variétés hyperboliques, Inven. Math. 114 (1993) 193-206. MR 94i:58158

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 53C12, 53C23, 57R30, 58F15, 58F18, 53C22, 57M99, 58F25

Retrieve articles in all journals with MSC (1991): 53C12, 53C23, 57R30, 58F15, 58F18, 53C22, 57M99, 58F25

Additional Information

Sérgio R. Fenley
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
Address at time of publication: Department of Mathematics, Washington University, St. Louis, Missouri 63130-4899

Keywords: Foliations, flows, hyperbolic 3-manifolds, geometric structures, asymptotic geometry, quasi-isometries
Received by editor(s): October 20, 1997
Received by editor(s) in revised form: March 5, 1998
Published electronically: April 26, 1999
Additional Notes: This research was partially supported by an NSF postdoctoral fellowship.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society