december 2021 Regular parallelisms on $\mathrm{PG}(3,\mathbb R)$ admitting a 2-torus action
Rainer Löwen, Günter F. Steinke
Bull. Belg. Math. Soc. Simon Stevin 28(2): 305-326 (december 2021). DOI: 10.36045/j.bbms.210114


A regular parallelism of real projective 3-space $\mathrm{PG}(3,\mathbb R)$ is an equivalence relation on the line space such that every class is equivalent to the set of 1-dimensional complex subspaces of $\mathbb C^2 = \mathbb R^4$. We shall assume that the set of classes is compact, and characterize those regular parallelisms that admit an action of a 2-dimensional torus group. We prove that there is a one-dimensional subtorus fixing every parallel class. From this property alone we deduce that the parallelism is a 2- or 3-dimensional regular parallelism in the sense of Betten and Riesinger [6]. If a 2-torus acts, then the parallelism can be described using a so-called generalized line star or \it gl star \rm which admits a 1-torus action. We also study examples of such parallelisms by constructing gl stars. In particular, we prove a claim which was presented in [6] with an incorrect proof. The present article continues a series of papers by the first author on parallelisms with large groups.


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Rainer Löwen. Günter F. Steinke. "Regular parallelisms on $\mathrm{PG}(3,\mathbb R)$ admitting a 2-torus action." Bull. Belg. Math. Soc. Simon Stevin 28 (2) 305 - 326, december 2021.


Published: december 2021
First available in Project Euclid: 23 December 2021

Digital Object Identifier: 10.36045/j.bbms.210114

Primary: 51A15 , 51H10 , 51M30 , 51M99

Keywords: automorphism group , generalized line star , regular parallelism , torus group

Rights: Copyright © 2021 The Belgian Mathematical Society


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Vol.28 • No. 2 • december 2021
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