  In translation surfaces of finite area (corresponding to holomorphic
differentials), directions of saddle connections are dense in the unit circle.
On the contrary, saddle connections are fewer in translation surfaces with
poles (corresponding to meromorphic differentials). The Cantor-Bendixson rank
of their set of directions is a measure of descriptive set-theoretic
complexity. Drawing on a previous work of David Aulicino, we prove a sharp
upper bound that depends only on the genus of the underlying topological
surface. The proof uses a new geometric lemma stating that in a sequence of
three nested invariant subsurfaces the genus of the third one is always bigger
than the genus of the first one.
