There is a useful family of projections
$ MCG(S) \to C(W) $
from the mapping class group of a surface S to curve complexes of subsurfaces W. Together they give a map to the product of curve complexes,
MCG(S) \to \prod_W C(W).
I will discuss a characterization of the image of this map in terms of simple inequalities. This characterization makes some simple “convex hull” constructions possible in MCG(S), and has applications to rigidity properties of the group. Joint work with J. Behrstock, B. Kleiner, and L. Mosher.