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

## TL;DR

This paper investigates the projection properties of Poisson cut-out sets in the Heisenberg group and the 3-sphere, revealing dimension relations and projection behaviors that extend classical Euclidean results to non-Euclidean geometries.

## Contribution

It establishes the Hausdorff dimension of projections in the Heisenberg group and the 3-sphere, demonstrating almost sure dimension preservation and the absence of exceptional directions in projections.

## Key findings

- Hausdorff dimension of vertical projections in Heisenberg group equals min{2, dim_H(E)}
- Projections in S^3 satisfy a strong Marstrand-type theorem with no exceptional directions
- Poisson cut-outs exhibit non-empty interior in projections when dimension exceeds 2

## Abstract

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