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

Frankel, Steven

Ann. Math. (2) 188, No. 1, 1-48 (2018).

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.

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

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