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

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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 be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then is not quasi-isometric, but suppose that 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 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.

**[An]**D. V. Anosov,*Geodesic flows on closed Riemann manifolds with negative curvature.*, Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969. MR**0242194****[An-Si]**D. V. Anosov and Ja. G. Sinaĭ,*Certain smooth ergodic systems*, Uspehi Mat. Nauk**22**(1967), no. 5 (137), 107–172 (Russian). MR**0224771****[As]**Daniel Asimov,*Round handles and non-singular Morse-Smale flows*, Ann. of Math. (2)**102**(1975), no. 1, 41–54. MR**0380883****[Ba]**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****[Be-Me]**Mladen Bestvina and Geoffrey Mess,*The boundary of negatively curved groups*, J. Amer. Math. Soc.**4**(1991), no. 3, 469–481. MR**1096169**, 10.1090/S0894-0347-1991-1096169-1**[Bl-Ca]**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****[Bo]**Francis Bonahon,*Bouts des variétés hyperboliques de dimension 3*, Ann. of Math. (2)**124**(1986), no. 1, 71–158 (French). MR**847953**, 10.2307/1971388**[Br]**Mark Brittenham,*Essential laminations in Seifert-fibered spaces*, Topology**32**(1993), no. 1, 61–85. MR**1204407**, 10.1016/0040-9383(93)90038-W**[BNR]**Mark Brittenham, Ramin Naimi, and Rachel Roberts,*Graph manifolds and taut foliations*, J. Differential Geom.**45**(1997), no. 3, 446–470. MR**1472884****[Can]**Alberto Candel,*Uniformization of surface laminations*, Ann. Sci. École Norm. Sup. (4)**26**(1993), no. 4, 489–516. MR**1235439****[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), no. 3, 219–244. MR**976690****[CLR1]**D. Cooper, D. D. Long, and A. W. Reid,*Bundles and finite foliations*, Invent. Math.**118**(1994), no. 2, 255–283. MR**1292113**, 10.1007/BF01231534**[CLR2]**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**, 10.1016/0040-9383(95)00066-6**[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]**Sérgio R. Fenley,*Asymptotic properties of depth one foliations in hyperbolic 3-manifolds*, J. Differential Geom.**36**(1992), no. 2, 269–313. MR**1180384****[Fe2]**Sérgio R. Fenley,*Quasi-isometric foliations*, Topology**31**(1992), no. 3, 667–676. MR**1174265**, 10.1016/0040-9383(92)90057-O**[Fe3]**Sérgio R. Fenley,*Anosov flows in 3-manifolds*, Ann. of Math. (2)**139**(1994), no. 1, 79–115. MR**1259365**, 10.2307/2946628**[Fe4]**Sérgio R. Fenley,*Quasigeodesic Anosov flows and homotopic properties of flow lines*, J. Differential Geom.**41**(1995), no. 2, 479–514. MR**1331975****[Fe5]**Sérgio R. Fenley,*One sided branching in Anosov foliations*, Comment. Math. Helv.**70**(1995), no. 2, 248–266. MR**1324629**, 10.1007/BF02566007**[Fe6]**Sérgio R. Fenley,*The structure of branching in Anosov flows of 3-manifolds*, Comment. Math. Helv.**73**(1998), no. 2, 259–297. MR**1611703**, 10.1007/s000140050055**[Fe7]**S. Fenley,*Surfaces transverse to pseudo-Anosov flows and virtual fibers in -manifolds*, to appear in Topology (1999).**[Fe-Mo]**S. Fenley and L. Mosher,*Quasigeodesic flows in hyperbolic -manifolds*, to appear in Topology.**[Fr-Wi]**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****[Ga1]**David Gabai,*Foliations and the topology of 3-manifolds*, J. Differential Geom.**18**(1983), no. 3, 445–503. MR**723813****[Ga2]**David Gabai,*Foliations and the topology of 3-manifolds. II*, J. Differential Geom.**26**(1987), no. 3, 461–478. MR**910017****[Ga3]**David Gabai,*Foliations and the topology of 3-manifolds. III*, J. Differential Geom.**26**(1987), no. 3, 479–536. MR**910018****[Ga-Oe]**David Gabai and Ulrich Oertel,*Essential laminations in 3-manifolds*, Ann. of Math. (2)**130**(1989), no. 1, 41–73. MR**1005607**, 10.2307/1971476**[Gh-Ha]**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****[Gr]**M. Gromov,*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, 10.1007/978-1-4613-9586-7_3**[He]**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****[Ja-Sh]**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**, 10.1090/memo/0220**[Le]**Gilbert Levitt,*Foliations and laminations on hyperbolic surfaces*, Topology**22**(1983), no. 2, 119–135. MR**683752**, 10.1016/0040-9383(83)90023-X**[Ma]**Albert Marden,*The geometry of finitely generated kleinian groups*, Ann. of Math. (2)**99**(1974), 383–462. MR**0349992****[Mor]**J. Morgan,*On Thurston's uniformization theorem for -dimensional manifolds*, in The Smith Conjecture, J. Morgan and H. Bass, eds., Academic Press, New York, 1984, 37-125. CMP**17:01****[Mo1]**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**, 10.1215/S0012-7094-92-06518-5**[Mo2]**Lee Mosher,*Dynamical systems and the homology norm of a 3-manifold. II*, Invent. Math.**107**(1992), no. 2, 243–281. MR**1144424**, 10.1007/BF01231890**[Mo3]**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****[Mo4]**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.**[No]**S. P. Novikov,*The topology of foliations*, Trudy Moskov. Mat. Obšč.**14**(1965), 248–278 (Russian). MR**0200938****[Pa]**Carlos Frederico Borges Palmeira,*Open manifolds foliated by planes*, Ann. Math. (2)**107**(1978), no. 1, 109–131. MR**0501018****[Pl]**J. F. Plante,*Foliations with measure preserving holonomy*, Ann. of Math. (2)**102**(1975), no. 2, 327–361. MR**0391125****[Ro]**Harold Rosenberg,*Foliations by planes*, Topology**7**(1968), 131–138. MR**0228011****[St]**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****[Sc]**Peter Scott,*The geometries of 3-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, 10.1112/blms/15.5.401**[Su]**Dennis Sullivan,*Cycles for the dynamical study of foliated manifolds and complex manifolds*, Invent. Math.**36**(1976), 225–255. MR**0433464****[Th1]**W. Thurston,*Foliations of -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 -manifolds II: Surface groups and -manifolds that fiber over the circle*, preprint.**[Th4]**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**, 10.1090/S0273-0979-1982-15003-0**[Ze]**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**, 10.1007/BF01232666

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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

DOI:
http://dx.doi.org/10.1090/S0894-0347-99-00304-5

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