  Translation surfaces with poles correspond to meromorphic differentials on
compact Riemann surfaces. They appear in compactifications of strata of the
moduli space of Abelian differentials and in the study of stability conditions.
Such structures have different geometrical and dynamical properties than usual
translation surfaces. In particular, they always have infinite area and can
have a finite number of saddle connections. We provide a combinatorial
characterization of the strata for which there can be an infinite number of
saddle connections and give lower and upper bounds for the number of saddle
connections and related quantities.
