Geometry & Topology

Pseudo-Anosov flows in toroidal manifolds

Thierry Barbot and Sérgio R Fenley

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We first prove rigidity results for pseudo-Anosov flows in prototypes of toroidal 3–manifolds: we show that a pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically equivalent to a geodesic flow and we show that a pseudo-Anosov flow in a solv manifold is topologically equivalent to a suspension Anosov flow. Then we study the interaction of a general pseudo-Anosov flow with possible Seifert fibered pieces in the torus decomposition: if the fiber is associated with a periodic orbit of the flow, we show that there is a standard and very simple form for the flow in the piece using Birkhoff annuli. This form is strongly connected with the topology of the Seifert piece. We also construct a large new class of examples in many graph manifolds, which is extremely general and flexible. We construct other new classes of examples, some of which are generalized pseudo-Anosov flows which have one-prong singularities and which show that the above results in Seifert fibered and solvable manifolds do not apply to one-prong pseudo-Anosov flows. Finally we also analyse immersed and embedded incompressible tori in optimal position with respect to a pseudo-Anosov flow.

Article information

Geom. Topol., Volume 17, Number 4 (2013), 1877-1954.

Received: 19 November 2011
Revised: 22 March 2013
Accepted: 21 February 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D50: Hyperbolic systems with singularities (billiards, etc.)
Secondary: 57M60: Group actions in low dimensions 57R30: Foliations; geometric theory

Pseudo-Anosov flows toroidal manifolds Seifert fibered spaces graph manifolds


Barbot, Thierry; Fenley, Sérgio R. Pseudo-Anosov flows in toroidal manifolds. Geom. Topol. 17 (2013), no. 4, 1877--1954. doi:10.2140/gt.2013.17.1877.

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  • D V Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Proceedings of the Steklov Institute of Mathematics 90, Amer. Math. Soc. (1969)
  • T Barbot, Caractérisation des flots d'Anosov en dimension $3$ par leurs feuilletages faibles, Ergodic Theory Dynam. Systems 15 (1995) 247–270
  • T Barbot, Mise en position optimale de tores par rapport à un flot d'Anosov, Comment. Math. Helv. 70 (1995) 113–160
  • T Barbot, Flots d'Anosov sur les variétés graphées au sens de Waldhausen, Ann. Inst. Fourier (Grenoble) 46 (1996) 1451–1517
  • T Barbot, Actions de groupes sur les $1$–variétés non séparées et feuilletages de codimension un, Ann. Fac. Sci. Toulouse Math. 7 (1998) 559–597
  • T Barbot, Generalizations of the Bonatti–Langevin example of Anosov flow and their classification up to topological equivalence, Comm. Anal. Geom. 6 (1998) 749–798
  • T Barbot, Plane affine geometry and Anosov flows, Ann. Sci. École Norm. Sup. 34 (2001) 871–889
  • T Barbot, De l'hyperbolique au globalement hyperbolique (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • C Bonatti, R Langevin, Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension, Ergodic Theory Dynam. Systems 14 (1994) 633–643
  • D Calegari, The geometry of $\mathbb R$–covered foliations, Geom. Topol. 4 (2000) 457–515
  • D Calegari, Foliations with one-sided branching, Geom. Dedicata 96 (2003) 1–53
  • D Calegari, Promoting essential laminations, Invent. Math. 166 (2006) 583–643
  • A Casson, D Jungreis, Convergence groups and Seifert fibered $3$–manifolds, Invent. Math. 118 (1994) 441–456
  • S R Fenley, Anosov flows in $3$–manifolds, Ann. of Math. 139 (1994) 79–115
  • S R Fenley, The structure of branching in Anosov flows of $3$–manifolds, Comment. Math. Helv. 73 (1998) 259–297
  • S R Fenley, Foliations with good geometry, J. Amer. Math. Soc. 12 (1999) 619–676
  • S R Fenley, Foliations, topology and geometry of $3$–manifolds: $\mathbb R$–covered foliations and transverse pseudo-Anosov flows, Comment. Math. Helv. 77 (2002) 415–490
  • S R Fenley, Pseudo–Anosov flows and incompressible tori, Geom. Dedicata 99 (2003) 61–102
  • S R Fenley, Laminar free hyperbolic $3$–manifolds, Comment. Math. Helv. 82 (2007) 247–321
  • S R Fenley, Geometry of foliations and flows, I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations, J. Differential Geom. 81 (2009) 1–89
  • S R Fenley, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry, Geom. Topol. 16 (2012) 1–110
  • S R Fenley, L Mosher, Quasigeodesic flows in hyperbolic $3$–manifolds, Topology 40 (2001) 503–537
  • J Franks, Anosov diffeomorphisms, from: “Global Analysis”, Amer. Math. Soc. (1970) 61–93
  • J Franks, B Williams, Anomalous Anosov flows, from: “Global theory of dynamical systems”, (Z Nitecki, C Robinson, editors), Lecture Notes in Math. 819, Springer, Berlin (1980) 158–174
  • D Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology 22 (1983) 299–303
  • D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447–510
  • D Gabai, W H Kazez, Group negative curvature for $3$–manifolds with genuine laminations, Geom. Topol. 2 (1998) 65–77
  • D Gabai, U Oertel, Essential laminations in $3$–manifolds, Ann. of Math. 130 (1989) 41–73
  • É Ghys, Flots d'Anosov sur les $3$–variétés fibrées en cercles, Ergodic Theory Dynam. Systems 4 (1984) 67–80
  • S Goodman, Dehn surgery on Anosov flows, from: “Geometric dynamics”, (J Palis, Jr, editor), Lecture Notes in Math. 1007, Springer, Berlin (1983) 300–307
  • M Handel, W P Thurston, Anosov flows on new three manifolds, Invent. Math. 59 (1980) 95–103
  • J Hempel, $3$–Manifolds, Ann. of Math. Studies 86, Princeton Univ. Press (1976)
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society (1980)
  • W Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, from: “Geometric topology”, (J C Cantrell, editor), Academic Press, New York (1979) 91–99
  • K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Mathematics 761, Springer, Berlin (1979)
  • J L Kelley, General topology, D. Van Nostrand Company, New York (1955)
  • L Mosher, Laminations and flows transverse to finite depth foliations Part I: Branched surfaces and dynamics, Part II in preparation Available at \setbox0\makeatletter\@url {\unhbox0
  • L Mosher, Dynamical systems and the homology norm of a $3$–manifold. I: Efficient intersection of surfaces and flows, Duke Math. J. 65 (1992) 449–500
  • L Mosher, Dynamical systems and the homology norm of a $3$–manifold, II, Invent. Math. 107 (1992) 243–281
  • G Perelman, The entropy formula for the Ricci flow and its geometric applications
  • G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
  • G Perelman, Ricci flow with surgery on three-manifolds
  • R Roberts, J Shareshian, M Stein, Infinitely many hyperbolic $3$–manifolds which contain no Reebless foliation, J. Amer. Math. Soc. 16 (2003) 639–679
  • R Roberts, M Stein, Group actions on order trees, Topology Appl. 115 (2001) 175–201
  • W P Thurston, Hyperbolic structures on $3$–manifolds II: Surface groups and $3$–manifolds that fiber over the circle, preprint
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417–431
  • W P Thurston, Three manifolds, foliations and circles, I, preprint (1997)
  • W P Thurston, Three manifolds, foliations and circles, II: The transverse asymptotic geometry of foliations, preprint (1998)
  • R Waller, Surfaces which are flow graphs, in preparation