  The connection between cutting sequences of geodesics on the modular surface
$\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$ and regular continued
fractions was established by Series, and Heersink expanded the cross-section of
the geodesic flow on the unit tangent bundle to the modular surface to describe
the Farey tent-map as a slowdown of the Gauss map for the regular continued
fractions. Boca and the author expanded the connection between cutting
sequences of geodesics on the modular surface $\Theta\backslash\mathbb{H}$ and
even continued fractions, which was previously established as a billiard flow
by Bauer and Lopes. We will similarly expand the cross-section of the geodesic
flow on this unit tangent bundle to describe the three-branch slowdown of the
even Farey map.
