Geodesic flow, left-handedness and templates

Abstract

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