Regular parallelisms on PG(3,R) admitting a 2-torus action

TL;DR

This paper characterizes regular parallelisms in real projective 3-space that admit a 2-torus group action, showing they are related to generalized line stars and extending previous work on parallelisms with large symmetry groups.

Contribution

It provides a complete characterization of regular parallelisms with 2-torus symmetry and introduces the concept of generalized line stars to describe such structures.

Findings

Existence of a 1-dimensional subtorus fixing each parallel class

Parallelisms with 2-torus action are 2- or 3-dimensional in Betten and Riesinger's sense

Construction of examples using generalized line stars with 1-torus actions

Abstract

A regular parallelism of real projective 3-space PG(3,R) is an equivalence relation on the line space such that every class is equivalent to the set of 1-dimensional complex subspaces of a 2-dimensional complex vector space. 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. If a 2-torus acts, then the parallelism can be described using a so-called generalized line star which admits a 1-torus action. We also study examples of such parallelisms by constructing generalized line stars. In particular, we prove a claim which was presented by Betten and Riesinger with an incorrect proof. The present article continues a series of papers by the first author on parallelisms with large groups.