DUFLOUX, L., & SUOMALA, V. (2022). Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere. Mathematical Proceedings of the Cambridge Philosophical Society, 172(1), 197-230. doi:10.1017/S0305004121000177

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

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Author: Dufloux, Laurent1; Suomala, Ville2
Organizations: 1Department of Mathematics and Statistics, P.O. Box 35, FI-40014, University of Jyväskylä, Finland
2Department of Mathematical Sciences, P.O. Box 8000, FI-90014, University of Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.5 MB)
Language: English
Published: Cambridge University Press, 2022
Publish Date: 2023-01-11
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# 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.

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Series: Mathematical proceedings of the Cambridge Philosophical Society
ISSN: 0305-0041
ISSN-E: 1469-8064
ISSN-L: 0305-0041
Volume: 172
Issue: 1
Pages: 197 - 230
DOI: 10.1017/s0305004121000177