Geodesic flow, left-handedness and templates

Abstract

We establish that for every hyperbolic orbifold of type (2; q;¥) 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. We also observe that the natural extension of the conjecture to arbitrary hyperbolic surfaces (with non-trivial homology) is false.