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

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