Cross sections for geodesic flows and alpha-continued fractions

We adjust Arnouxs coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the alpha-continued fractions, for each $alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced alpha-variants.