University of Oulu

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
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Language: English
Published: Cambridge University Press, 2022
Publish Date: 2023-01-11


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
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: We acknowledge support from the following sources: Academy of Finland via the Centre of Excellence in Analysis and Dynamics research and the research project Geometry of subRiemannian Groups (#288501), the European Research Council via the ERC Starting Grant #713998 GeoMeG ‘Geometry of Metric Groups’, the Institute Mittag-Leffler via the Fractal Geometry and Dynamics research program.
Copyright information: © The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society. This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society This version is free to view and download for private research and study only. Not for re-distribution or re-use.