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.


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
Full Text: DOI arXiv


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