Projections of Poisson cutouts in the Heisenberg group and the visual 3sphere 

Author:  Dufloux, Laurent^{1}; Suomala, Ville^{2} 
Organizations: 
^{1}Department of Mathematics and Statistics, P.O. Box 35, FI40014, University of Jyväskylä, Finland ^{2}Department of Mathematical Sciences, P.O. Box 8000, FI90014, University of Oulu, Finland 
Format:  article 
Version:  accepted version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.5 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife202301112355 
Language:  English 
Published: 
Cambridge University Press,
2022

Publish Date:  20230111 
Description: 
AbstractWe study projectional properties of Poisson cutout sets \(E\) in nonEuclidean 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 nonempty 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 onepoint compactification of the Heisenberg group, that is, the 3sphere \(\mathbf{S}^3\) 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 cutouts in \(\mathbf{S}^3\) satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions. see all

Series: 
Mathematical proceedings of the Cambridge Philosophical Society 
ISSN:  03050041 
ISSNE:  14698064 
ISSNL:  03050041 
Volume:  172 
Issue:  1 
Pages:  197  230 
DOI:  10.1017/s0305004121000177 
OADOI:  https://oadoi.org/10.1017/s0305004121000177 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
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 MittagLeffler 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 http://doi.org/10.1017/s0305004121000177. This version is free to view and download for private research and study only. Not for redistribution or reuse. 