Projection and slicing theorems in Heisenberg groups

We study the behavior of the Hausdorff dimension of sets in the Heisenberg group Hn, n∈N, with a sub-Riemannian metric under projections onto horizontal and vertical subgroups, and under slicing by translates of vertical subgroups. We formulate almost sure statements in terms of a natural measure on the Grassmannian of isotropic subspaces.