We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group
\[\mathbb{H} = \mathbb{C} \times \mathbb{R}\]
, endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection
\[\pi (E)\]
(projection along the center of
\[\mathbb{H}\]
) almost surely equals
\[\min \{ 2,{\dim _\operatorname{H} }(E)\} \]
and that
\[\pi (E)\]
has non-empty interior if
\[{\dim _{\text{H}}}(E) > 2\]
. 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
\[{\dim _{\text{H}}}(E)\]
.
We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere
\[{{\text{S}}^3}\]
endowed with the visual metric d obtained by identifying
\[{{\text{S}}^3}\]
with the boundary of the complex hyperbolic plane. In
\[{{\text{S}}^3}\]
, 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
\[{{\text{S}}^3}\]
satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.