  We study the behavior of Sobolev mappings defined on the Heisenberg groups
with respect to a foliation by left cosets of a horizontal homogeneous
subgroup. We quantitatively estimate, in terms of Euclidean Hausdorff
dimension, the size of the set of cosets that are mapped onto sets of high
dimension. The proof of our main result combines ideas of Gehring and Mostow
about the absolute continuity of quasiconformal mappings with Mattila's
projection and slicing machinery.
