Transversality of isotropic projections, unrectifiability and Heisenberg groups

We show that the family of $m$-dimensional isotropic projections in $\R^{2n}$ is transversal. As an application we show that the Besicovitch-Federer projection theorem holds for isotropic projections. We also use transversality to obtain almost sure estimates on the Hausdorff dimension of isotropic projections of subsets $E \subset \R^{2n}$. These results may also be applied to gain information on the horizontal projections of the Heisenberg group \H^n.