Suspension flows are quasigeodesic. (English) Zbl 1125.53068

Hoffoss, Diane

J. Differ. Geom. 76, No. 2, 215-248 (2007).

Let \(F\) be a hyperbolic surface, and let \(M:= F\times [0,1]/(x,1)\equiv (\psi(x),0)\) be the hyperbolic 3-manifold fibering over the circle, which is the suspension of \(F\) obtained by gluing \(F\times\{1\}\) to \(F\times\{0\}\) by means of a pseudo-Anosov monodromy map \(\psi\). The suspension flow on \(M\) is obtained by projecting the lines \(\{x\}\times\mathbb{R}\) on \(M\), from the cover \(F\times\mathbb{R}\).
The main result is that this suspension flow can be isotoped to be uniformly quasigeodesic, meaning that the flow lines, lifted to hyperbolic space, are bi-Lipschitz embeddings of \(\mathbb{R}\).
This result extends results by Cannon and Thurston and by Zeghib, which dealt with compact fibering 3-manifold \(M\), by allowing \(M\) to have a finite number of cusps. (Finsler) singular Solv metric plays a noteworthy role in the proof.

Reviewer: Jacques Franchi (Strasbourg)

MSC:

53D25	Geodesic flows in symplectic geometry and contact geometry
53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53C12	Foliations (differential geometric aspects)
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
53C60	Global differential geometry of Finsler spaces and generalizations (areal metrics)

Keywords:

suspension flow; quasigeodesics; cusped hyperbolic 3-manifold; Solv metric