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**Coarse hyperbolicity and closed orbits for quasigeodesic flows.**
*(English)*
Zbl 1397.57036

The main result of this paper is what the author calls the Closed Orbits Theorem, which establishes that every quasigeodesic flow on a closed hyperbolic 3-manifold contains a closed orbit, as conjectured by Calegari.

A flow \(\Phi:\mathbb{R}\times M\to M\) defined on a manifold is quasigeodesic if each orbit lifts to a quasigeodesic in the universal cover \(\widetilde{M}\), i.e., each lifted orbit admits a parametrization \(\gamma:\mathbb{R}\to \widetilde{M}\) such that \[ \frac{1}{k}d(\gamma(s),\gamma(t))-\varepsilon\leq|s-t|\leq kd(\gamma(x),\gamma(y))+\varepsilon \] for constants \(k>0\), \(\varepsilon>0\) that may depend on the orbit.

To prove the existence of closed orbits, the author, instead of addressing directly the 3-dimensional problem, reduces the problem to a 2-dimensional one. In particular, he studies the space of orbits of the lifted flow \(\widetilde{\Phi}\) to the universal cover \(\widetilde{M}\simeq \mathbb{H}^3\). This space of orbits is a topological plane \(P\) that carries a natural action of \(\pi_1(M)\). The properties of recurrence and periodicity of the flow \(\Phi\) can be seen in terms of this action and, in particular, each closed trajectory of \(\Phi\) corresponds to a point in \(P\) which is fixed by a non-trivial element of \(\pi_1(M)\). To find such an element, the author studies a pair of decompositions \(\mathcal{D}^{\pm}\) of \(P\), induced by the orbits of \(\widetilde{\Phi}\), and their extensions to the compactification \(\mathbf{P}=P\sqcup S^1_ u\) of \(P\) by Calegari’s universal circle, which is a topological disk.

At the end of the paper the author presents some questions and conjectures.

A flow \(\Phi:\mathbb{R}\times M\to M\) defined on a manifold is quasigeodesic if each orbit lifts to a quasigeodesic in the universal cover \(\widetilde{M}\), i.e., each lifted orbit admits a parametrization \(\gamma:\mathbb{R}\to \widetilde{M}\) such that \[ \frac{1}{k}d(\gamma(s),\gamma(t))-\varepsilon\leq|s-t|\leq kd(\gamma(x),\gamma(y))+\varepsilon \] for constants \(k>0\), \(\varepsilon>0\) that may depend on the orbit.

To prove the existence of closed orbits, the author, instead of addressing directly the 3-dimensional problem, reduces the problem to a 2-dimensional one. In particular, he studies the space of orbits of the lifted flow \(\widetilde{\Phi}\) to the universal cover \(\widetilde{M}\simeq \mathbb{H}^3\). This space of orbits is a topological plane \(P\) that carries a natural action of \(\pi_1(M)\). The properties of recurrence and periodicity of the flow \(\Phi\) can be seen in terms of this action and, in particular, each closed trajectory of \(\Phi\) corresponds to a point in \(P\) which is fixed by a non-trivial element of \(\pi_1(M)\). To find such an element, the author studies a pair of decompositions \(\mathcal{D}^{\pm}\) of \(P\), induced by the orbits of \(\widetilde{\Phi}\), and their extensions to the compactification \(\mathbf{P}=P\sqcup S^1_ u\) of \(P\) by Calegari’s universal circle, which is a topological disk.

At the end of the paper the author presents some questions and conjectures.

Reviewer: Héctor Barge (Madrid)

### MSC:

57M60 | Group actions on manifolds and cell complexes in low dimensions |

57M50 | General geometric structures on low-dimensional manifolds |

37C27 | Periodic orbits of vector fields and flows |

### Keywords:

quasigeodesic flows; pseudo-Anosov flows; closed orbits; periodic orbits; hyperbolic dynamics; hyperbolic manifolds### References:

[1] | Anosov, D. V., Ergodic properties of geodesic flows on closed {R}iemannian manifolds of negative curvature, Dokl. Akad. Nauk SSSR. Doklady Akademii Nauk SSSR, 151, 1250-1252, (1963) · Zbl 0135.40402 |

[2] | Auslander, Joseph, Minimal Flows and their Extensions, North-Holland Math. Stud., 153, xii+265 pp., (1988) · Zbl 0654.54027 |

[3] | Bridson, Martin R.; Haefliger, Andr\'e, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss., 319, xxii+643 pp., (1999) · Zbl 0988.53001 |

[4] | Calegari, Danny, Universal circles for quasigeodesic flows, Geom. Topol.. Geometry and Topology, 10, 2271-2298, (2006) · Zbl 1129.57032 |

[5] | Cannon, James W.; Thurston, William P., Group invariant {P}eano curves, Geom. Topol.. Geometry & Topology, 11, 1315-1355, (2007) · Zbl 1136.57009 |

[6] | Dawson, Robert J. {\relax{MacG}}., Unsolved {P}roblems: {P}aradoxical {C}onnections, Amer. Math. Monthly. American Mathematical Monthly, 96, 31-33, (1989) · Zbl 0687.54024 |

[7] | Fenley, S\'ergio, Ideal boundaries of pseudo-{A}nosov flows and uniform convergence groups with connections and applications to large scale geometry, Geom. Topol.. Geometry & Topology, 16, 1-110, (2012) · Zbl 1279.37026 |

[8] | Fenley, S\'ergio; Mosher, Lee, Quasigeodesic flows in hyperbolic 3-manifolds, Topology. Topology. An International Journal of Mathematics, 40, 503-537, (2001) · Zbl 0990.53040 |

[9] | Frankel, Steven, Quasigeodesic flows and {M}\"obius-like groups, J. Differential Geom.. Journal of Differential Geometry, 93, 401-429, (2013) · Zbl 1279.53063 |

[10] | Frankel, Steven, Quasigeodesic flows and sphere-filling curves, Geom. Topol.. Geometry & Topology, 19, 1249-1262, (2015) · Zbl 1327.57021 |

[11] | Franks, John, A new proof of the {B}rouwer plane translation theorem, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 12, 217-226, (1992) · Zbl 0767.58025 |

[12] | Gabai, David, Foliations and the topology of {\(3\)}-manifolds, J. Differential Geom.. Journal of Differential Geometry, 18, 445-503, (1983) · Zbl 0533.57013 |

[13] | Gromov, M., Hyperbolic groups. Essays in {G}roup {T}heory, Math. Sci. Res. Inst. Publ., 8, 75-263, (1987) |

[14] | Hocking, John G.; Young, Gail S., Topology, ix+374 pp., (1961) · Zbl 0718.55001 |

[15] | Kapovich, Michael, Hyperbolic Manifolds and Discrete Groups, Progr. Math., 183, xxvi+467 pp., (2001) · Zbl 0958.57001 |

[16] | Kova\v{c}evi\'{c}, Nata\v{s}a, Examples of {M}\`“‘obius-like groups which are not {M}\'”’{o}bius groups, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 351, 4823-4835, (1999) · Zbl 0934.57039 |

[17] | Kuperberg, Greg, A volume-preserving counterexample to the {S}eifert conjecture, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 71, 70-97, (1996) · Zbl 0859.57017 |

[18] | Kuperberg, Krystyna, A smooth counterexample to the {S}eifert conjecture, Ann. of Math. (2). Annals of Mathematics. Second Series, 140, 723-732, (1994) · Zbl 0856.57024 |

[19] | Kuratowski, K., Topology. {V}ol. {I}, xx+560 pp., (1966) · Zbl 0158.40901 |

[20] | Kuratowski, K., Topology. {V}ol. {II}, new edition, revised and augmented; translated from the French by A. Kirkor, xiv+608 pp., (1968) |

[21] | Mangum, Brian S., Incompressible surfaces and pseudo-{A}nosov flows, Topology Appl.. Topology and its Applications, 87, 29-51, (1998) · Zbl 0927.57009 |

[22] | Montgomery, Deane; Zippin, Leo, Translation {G}roups of {T}hree-{S}pace, Amer. J. Math.. American Journal of Mathematics, 59, 121-128, (1937) · Zbl 0016.10303 |

[23] | Moore, R. L., Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 27, 416-428, (1925) · JFM 51.0464.03 |

[24] | Rechtman, Ana, Existence of periodic orbits for geodesible vector fields on closed 3-manifolds, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 30, 1817-1841, (2010) · Zbl 1214.37015 |

[25] | Schweitzer, Paul A., Counterexamples to the {S}eifert conjecture and opening closed leaves of foliations, Ann. of Math. (2). Annals of Mathematics. Second Series, 100, 386-400, (1974) · Zbl 0295.57010 |

[26] | Seifert, Herbert, Closed integral curves in {\(3\)}-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc.. Proceedings of the American Mathematical Society, 1, 287-302, (1950) · Zbl 0039.40002 |

[27] | Taubes, Clifford Henry, The {S}eiberg-{W}itten equations and the {W}einstein conjecture, Geom. Topol.. Geometry & Topology, 11, 2117-2202, (2007) · Zbl 1135.57015 |

[28] | Thurston, W. P., Geometry and topology of 3-manifolds, (1979) |

[29] | Wilder, Raymond Louis, Topology of Manifolds, Amer. Math. Soc. Colloq. Publ., 32, xiii+403 pp., (1979) · Zbl 0511.57001 |

[30] | Zeghib, A., Sur les feuilletages g\'eod\'esiques continus des vari\'et\'es hyperboliques, Invent. Math.. Inventiones Mathematicae, 114, 193-206, (1993) · Zbl 0789.57019 |

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