Genus one Birkhoff sections for geodesic flows on orbifolds

Abstract

For $\mathcal{O}$ a hyperbolic orientable 2-orbifold of genus $g$ with at most $2g+6$ conic points, we prove that the geodesic flow on the unitary tangent bundle${\mathrm{T}}^1\mathcal{O}$ admits a Birkhoff section whose genus is one. Together with a result of Minakawa, this implies that this flow is almost equivalent to the suspension flow of the $(\begin{array}{c}2\&1\\ 1\&1\end{array})$-map on the torus.