*Preprint*

**Inserted:** 10 feb 2020

**Last Updated:** 10 feb 2020

**Year:** 2018

**Abstract:**

We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group $\mathcal{H}=\mathbf{C}\times\mathbf{R}$, endowed with the Koranyi metric, we show that the Hausdorff dimension of the vertical projection $\pi(E)$ (projection along the center of $\mathcal{H}$) almost surely equals $\min\{2,\dim(E)\}$ and that $\pi(E)$ has non-empty interior if $\dim(E)>2$. 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 $\dim (E)$. 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.

**Tags:**
GeoMeG

**Keywords:**
Hausdorff dimension, Heisenberg group, Koranyi metric