Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

Abstract

Abstract We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group $\mathbb{H} =\mathbb{C}×\mathbb{R}$, endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection $π(E)$ (projection along the center of $\mathbb{H}$) almost surely equals $\textrm{min}\{2, \textrm{dim}_{\textrm{H}}(E)\}$ and that $π(E)$ has non-empty interior if $\textrm{dim}_{\textrm{H}}(E) > 2$. As a corollary, this allows us to determine the Hausdorff dimension of $E$ with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension $\textrm{dim}_{\textrm{H}}(E)$. We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere $\mathbf{S}^3$ endowed with the visual metric $d$ obtained by identifying $\mathbf{S}^3$ with the boundary of the complex hyperbolic plane. In $\mathbf{S}^3$, we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in $\mathbf{S}^3$ satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions