With Calta and Kraaikamp we studied alpha-type families of continued fraction-like interval maps for each of a countably infinite collection of Fuchsian triangle groups. We showed that the measure theoretic entropy defines a continuous function of the parameter for each family. We introduced a notion of `first expansive power' of an interval map and conjectured that the first expansive power for each of our maps defines a system whose natural extension is given by a cross section to the geodesic flow on the unit tangent bundle of the hyperbolic surface uniformized by the corresponding triangle group. I will present a proof of the conjecture, after sketching background and motivation.