  We prove upper and lower bounds for the number of lines in general position
that are rich in a Cartesian product point set. This disproves a conjecture of
Solymosi and improves work of Elekes, Borenstein and Croot, and Amirkhanyan,
Bush, Croot, and Pryby.
  The upper bounds are based on a version of the asymmetric
Balog-Szemeredi-Gowers theorem for group actions combined with product theorems
for the affine group. The lower bounds are based on a connection between rich
lines in Cartesian product sets and amenability (or expanding families of
graphs in the finite field case).
  As an application of our upper bounds for rich lines in grids, we give a
geometric proof of the asymmetric sum-product estimates of Bourgain and
Shkredov.
