Geodesic flow, left-handedness, and templates

TL;DR

This paper proves that geodesic flows on certain hyperbolic orbifolds and flat tori are left-handed, leading to fibered links from periodic orbits, but shows the conjecture does not extend to all hyperbolic surfaces.

Contribution

It establishes left-handedness of geodesic flow for specific orbifolds and tori, and discusses limitations of extending the conjecture to all hyperbolic surfaces.

Findings

Geodesic flow on (2, q, ∞) orbifolds is left-handed.

Periodic orbit links bound Birkhoff sections and are fibered.

The conjecture does not hold for all hyperbolic surfaces.

Abstract

We establish that, for every hyperbolic orbifold of type (2, q, $\infty$) and for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent bundle is left-handed. This implies that the link formed by every collection of periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii) is a fibered link. We also prove similar results for the torus with any flat metric. Besides, we observe that the natural extension of the conjecture to arbitrary hyperbolic surfaces (with non-trivial homology) is false.