## Foliations with good geometry

HTML articles powered by AMS MathViewer

- by Sérgio R. Fenley PDF
- J. Amer. Math. Soc.
**12**(1999), 619-676 Request permission

## 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

- D. V. Anosov,
*Geodesic flows on closed Riemann manifolds with negative curvature.*, Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR**0242194** - D. V. Anosov and Ja. G. Sinaĭ,
*Certain smooth ergodic systems*, Uspehi Mat. Nauk**22**(1967), no. 5 (137), 107–172 (Russian). MR**0224771** - Daniel Asimov,
*Round handles and non-singular Morse-Smale flows*, Ann. of Math. (2)**102**(1975), no. 1, 41–54. MR**380883**, DOI 10.2307/1970972 - Thierry Barbot,
*Flots d’Anosov sur les variétés graphées au sens de Waldhausen*, Ann. Inst. Fourier (Grenoble)**46**(1996), no. 5, 1451–1517 (French, with English and French summaries). MR**1427133**, DOI 10.5802/aif.1556 - Mladen Bestvina and Geoffrey Mess,
*The boundary of negatively curved groups*, J. Amer. Math. Soc.**4**(1991), no. 3, 469–481. MR**1096169**, DOI 10.1090/S0894-0347-1991-1096169-1 - Andrew J. Casson and Steven A. Bleiler,
*Automorphisms of surfaces after Nielsen and Thurston*, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR**964685**, DOI 10.1017/CBO9780511623912 - Francis Bonahon,
*Bouts des variétés hyperboliques de dimension $3$*, Ann. of Math. (2)**124**(1986), no. 1, 71–158 (French). MR**847953**, DOI 10.2307/1971388 - Mark Brittenham,
*Essential laminations in Seifert-fibered spaces*, Topology**32**(1993), no. 1, 61–85. MR**1204407**, DOI 10.1016/0040-9383(93)90038-W - Mark Brittenham, Ramin Naimi, and Rachel Roberts,
*Graph manifolds and taut foliations*, J. Differential Geom.**45**(1997), no. 3, 446–470. MR**1472884** - Alberto Candel,
*Uniformization of surface laminations*, Ann. Sci. École Norm. Sup. (4)**26**(1993), no. 4, 489–516. MR**1235439**, DOI 10.24033/asens.1678 - J. Cannon and W. Thurston,
*Group invariant Peano curves*, to appear. - J. Cantwell and L. Conlon,
*Smoothability of proper foliations*, Ann. Inst. Fourier (Grenoble)**38**(1988), no. 3, 219–244. MR**976690**, DOI 10.5802/aif.1146 - D. Cooper, D. D. Long, and A. W. Reid,
*Bundles and finite foliations*, Invent. Math.**118**(1994), no. 2, 255–283. MR**1292113**, DOI 10.1007/BF01231534 - D. Cooper, D. D. Long, and A. W. Reid,
*Finite foliations and similarity interval exchange maps*, Topology**36**(1997), no. 1, 209–227. MR**1410472**, DOI 10.1016/0040-9383(95)00066-6 - 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). - Sérgio R. Fenley,
*Asymptotic properties of depth one foliations in hyperbolic $3$-manifolds*, J. Differential Geom.**36**(1992), no. 2, 269–313. MR**1180384** - Sérgio R. Fenley,
*Quasi-isometric foliations*, Topology**31**(1992), no. 3, 667–676. MR**1174265**, DOI 10.1016/0040-9383(92)90057-O - Sérgio R. Fenley,
*Anosov flows in $3$-manifolds*, Ann. of Math. (2)**139**(1994), no. 1, 79–115. MR**1259365**, DOI 10.2307/2946628 - Sérgio R. Fenley,
*Quasigeodesic Anosov flows and homotopic properties of flow lines*, J. Differential Geom.**41**(1995), no. 2, 479–514. MR**1331975** - Sérgio R. Fenley,
*One sided branching in Anosov foliations*, Comment. Math. Helv.**70**(1995), no. 2, 248–266. MR**1324629**, DOI 10.1007/BF02566007 - Sérgio R. Fenley,
*The structure of branching in Anosov flows of $3$-manifolds*, Comment. Math. Helv.**73**(1998), no. 2, 259–297. MR**1611703**, DOI 10.1007/s000140050055 - S. Fenley,
*Surfaces transverse to pseudo-Anosov flows and virtual fibers in $3$-manifolds*, to appear in Topology (1999). - S. Fenley and L. Mosher,
*Quasigeodesic flows in hyperbolic $3$-manifolds*, to appear in Topology. - John Franks and Bob Williams,
*Anomalous Anosov flows*, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 158–174. MR**591182** - David Gabai,
*Foliations and the topology of $3$-manifolds*, J. Differential Geom.**18**(1983), no. 3, 445–503. MR**723813** - David Gabai,
*Foliations and the topology of $3$-manifolds. II*, J. Differential Geom.**26**(1987), no. 3, 461–478. MR**910017** - David Gabai,
*Foliations and the topology of $3$-manifolds. II*, J. Differential Geom.**26**(1987), no. 3, 461–478. MR**910017** - David Gabai and Ulrich Oertel,
*Essential laminations in $3$-manifolds*, Ann. of Math. (2)**130**(1989), no. 1, 41–73. MR**1005607**, DOI 10.2307/1971476 - É. Ghys and P. de la Harpe (eds.),
*Sur les groupes hyperboliques d’après Mikhael Gromov*, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR**1086648**, DOI 10.1007/978-1-4684-9167-8 - M. Gromov,
*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, DOI 10.1007/978-1-4613-9586-7_{3} - John Hempel,
*$3$-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619** - 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 - Gilbert Levitt,
*Foliations and laminations on hyperbolic surfaces*, Topology**22**(1983), no. 2, 119–135. MR**683752**, DOI 10.1016/0040-9383(83)90023-X - Albert Marden,
*The geometry of finitely generated kleinian groups*, Ann. of Math. (2)**99**(1974), 383–462. MR**349992**, DOI 10.2307/1971059 - 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. - Lee Mosher,
*Dynamical systems and the homology norm of a $3$-manifold. I. Efficient intersection of surfaces and flows*, Duke Math. J.**65**(1992), no. 3, 449–500. MR**1154179**, DOI 10.1215/S0012-7094-92-06518-5 - Lee Mosher,
*Dynamical systems and the homology norm of a $3$-manifold. I. Efficient intersection of surfaces and flows*, Duke Math. J.**65**(1992), no. 3, 449–500. MR**1154179**, DOI 10.1215/S0012-7094-92-06518-5 - Lee Mosher,
*Examples of quasi-geodesic flows on hyperbolic $3$-manifolds*, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 227–241. MR**1184414** - L. Mosher,
*Laminations and flows transverse to finite depth foliations*, manuscript available from http://newark.rutgers.edu:80/ mosher/, Part I: Branched surfaces and dynamics, Part II in preparation. - S. P. Novikov,
*The topology of foliations*, Trudy Moskov. Mat. Obšč.**14**(1965), 248–278 (Russian). MR**0200938** - Carlos Frederico Borges Palmeira,
*Open manifolds foliated by planes*, Ann. of Math. (2)**107**(1978), no. 1, 109–131. MR**501018**, DOI 10.2307/1971256 - J. F. Plante,
*Foliations with measure preserving holonomy*, Ann. of Math. (2)**102**(1975), no. 2, 327–361. MR**391125**, DOI 10.2307/1971034 - Harold Rosenberg,
*Foliations by planes*, Topology**7**(1968), 131–138. MR**228011**, DOI 10.1016/0040-9383(68)90021-9 - Kurt Strebel,
*Quadratic differentials*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR**743423**, DOI 10.1007/978-3-662-02414-0 - Peter Scott,
*The geometries of $3$-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, DOI 10.1112/blms/15.5.401 - Dennis Sullivan,
*Cycles for the dynamical study of foliated manifolds and complex manifolds*, Invent. Math.**36**(1976), 225–255. MR**433464**, DOI 10.1007/BF01390011 - W. Thurston,
*Foliations of $3$-manifolds which are Circle Bundles*, Thesis, University of California, Berkeley, 1972. - W. Thurston,
*The geometry and topology of 3-manifolds*, Princeton University Lecture Notes, 1982. - W. Thurston,
*Hyperbolic structures on $3$-manifolds II: Surface groups and $3$-manifolds that fiber over the circle*, preprint. - William P. Thurston,
*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 3, 357–381. MR**648524**, DOI 10.1090/S0273-0979-1982-15003-0 - A. Zeghib,
*Sur les feuilletages géodésiques continus des variétés hyperboliques*, Invent. Math.**114**(1993), no. 1, 193–206 (French, with English and French summaries). MR**1235023**, DOI 10.1007/BF01232666

## 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
- Email: fenley@math.princeton.edu, fenley@math.wustl.edu
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**12**(1999), 619-676 - MSC (1991): Primary 53C12, 53C23, 57R30, 58F15, 58F18; Secondary 53C22, 57M99, 58F25
- DOI: https://doi.org/10.1090/S0894-0347-99-00304-5
- MathSciNet review: 1674739