  We determine five extremal effective rays of the four-dimensional cone of
effective surfaces on the toroidal compactification $\overline{\mathcal A}_3$
of the moduli space ${\mathcal A}_3$ of complex principally polarized abelian
threefolds, and we conjecture that the cone of effective surfaces is generated
by these surfaces. As the surfaces we define can be defined in any genus $g\ge
3$, we further conjecture that they generate the cone of effective surfaces on
the perfect cone toroidal compactification of ${\mathcal A}_g$ for any $g\ge
3$.
