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.

中文翻译：

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海森堡群和视觉 3 球中泊松切口的投影

我们研究泊松剪裁集的投影特性乙在非欧几里得空间。在第一海森堡集团\[\mathbb{H} = \mathbb{C} \times \mathbb{R}\]，赋予 Korányi 度量，我们证明了垂直投影的 Hausdorff 维数\[\pi (E)\]（沿中心的投影\[\mathbb{H}\]) 几乎肯定等于\[\min \{ 2,{\dim _\运营商名{H} }(E)\} \]然后\[\pi (E)\]如果有非空内部\[{\dim _{\text{H}}}(E) > 2\]. 作为推论，这使我们能够确定乙关于欧几里得度量的海森堡豪斯多夫维数\[{\dim _{\text{H}}}(E)\]. 我们还研究海森堡群的一点紧致化中的投影，即 3 球体\[{{\文本{S}}^3}\]赋予视觉度量d通过识别获得\[{{\文本{S}}^3}\]与复双曲平面的边界。在\[{{\文本{S}}^3}\]，我们证明了一个投影结果同时适用于所有径向投影（沿着所谓的“链”的投影）。这表明泊松切口在\[{{\文本{S}}^3}\]满足马斯特兰德投影定理的强版本，没有任何异常方向。