Transversality of isotropic projections, unrectifiability, and Heisenberg groups

Abstract

We show that the family of $m$-dimensional isotropic projections in ${\mathbb{R}}^{2n}$ 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 $E\subset {\mathbb{R}}^{2n}$. These results may also be applied to gain information on the horizontal projections of the Heisenberg group ${\mathbb{H}}^n$.