Geodesic flow, left-handedness, and templates

Pierre Dehornoy (1)

1 Mathematisches Institut [Bern]

We establish that, for every hyperbolic orbifolds 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. These results support a conjecture of Ghys that these properties hold for every 2-dimensional hyperbolic orbifold that is a homology sphere. 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.

Mots clés: geodesic; knot; template; linking number

Domaines: Topologie géométrique [math.GT]; Systèmes dynamiques [math.DS]

Pierre Dehornoy. Geodesic flow, left-handedness, and templates. 2012. ⟨hal-00655422v2⟩