[Paper Review] Rotational spreads and rotational parallelisms and oriented parallelisms of PG(3,R)

Rainer Löwen|arXiv (Cornell University)|Apr 20, 2018

Finite Group Theory Research10 references3 citations

TL;DR

This paper introduces oriented parallelisms of lines in PG(3,ℝ), a new class of topological parallelisms arising from rotational spreads that are not derived from ordinary parallelisms. It shows that while ordinary parallelisms from rotational Betten spreads are uniquely determined by their spread and SO₃ℝ action, oriented parallelisms yield far more non-isomorphic structures due to distinct group actions, with automorphism groups always isomorphic to SO₃ℝ, even when the underlying spread has larger symmetry.

ABSTRACT

We introduce topological parallelisms of oriented lines (briefly called oriented parallelisms). Every topological parallelism (of lines) on PG(3,R) gives rise to a parallelism of oriented lines, but we show that even the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way. In fact we determine all parallelisms of both types that admit a reducible SO(3)-action. Only the Clifford parallelism admits a larger group Surprisingly, it turns out that there are far more oriented parallelisms of this kind than ordinary parallelisms. In particular, we show that the SO(3)-invariant ordinary parallelisms constructed by Betten and Riesinger are the only ones compatible with this group action. We also study the rotational Betten spreads used in this construction and their automorphisms. The automorphism group of the (oriented) parallelisms is always SO(3), no matter how large the automorphism group of the non-regular spread is. There are far more isomorphism types of parallelisms than types of spreads.

Motivation & Objective

• To define and study oriented parallelisms of lines in PG(3,ℝ), a new class of topological parallelisms distinct from ordinary parallelisms.
• To investigate the automorphism groups of such oriented parallelisms and compare them with those of ordinary parallelisms.
• To determine when rotational spreads—especially acentric families of reguli—give rise to topological parallelisms with maximal SO₃ℝ symmetry.
• To clarify the relationship between isomorphism types of parallelisms and the choice of rotation group (SO₃ℝ) applied to a given spread.
• To show that the isomorphism type of the resulting parallelism depends not only on the spread but also on the specific SO₃ℝ action used, even when the spread’s automorphism group is small.

Proposed method

• Construct oriented parallelisms by applying a reducible SO₃ℝ-action to rotational spreads, including both concentric (Betten) and acentric families of reguli.
• Use the topological structure of the space of oriented lines as a double cover of the space of unoriented lines to define oriented parallelisms.
• Analyze the automorphism group of the resulting parallelisms using compactness and group-theoretic arguments, showing it is always SO₃ℝ.
• Prove that non-Clifford parallelisms with SO₃ℝ as automorphism group are uniquely characterized by their group action, ruling out larger automorphism groups.
• Demonstrate that different SO₃ℝ-conjugates of the same spread yield non-isomorphic parallelisms by showing the automorphism group of the parallelism is precisely the rotation group.
• Use the normalizer of the SO₂ℝ-action within AGL₃ℝ to generate uncountably many distinct SO₃ℝ-conjugates, each yielding a distinct parallelism.

Experimental results

Research questions

• RQ1 Can oriented parallelisms be constructed from acentric families of reguli, and do they yield new parallelism types not obtainable from ordinary parallelisms?
• RQ2 Why do oriented parallelisms admit far more non-isomorphic structures than ordinary parallelisms when the same spread is rotated by different SO₃ℝ actions?
• RQ3 What is the automorphism group of a parallelism constructed via a reducible SO₃ℝ-action on a rotational spread, and how does it relate to the spread’s own automorphism group?
• RQ4 Under what conditions is the isomorphism type of a parallelism determined solely by the spread, and when does it depend on the rotation group used?
• RQ5 Can the same rotational spread generate uncountably many non-isomorphic oriented parallelisms via different SO₃ℝ-conjugate actions?

Key findings

• Oriented parallelisms can be constructed from both concentric (Betten) and acentric families of reguli, extending the classical construction of ordinary parallelisms.
• The automorphism group of every non-Clifford parallelism constructed via a reducible SO₃ℝ-action is exactly SO₃ℝ, regardless of the spread’s own automorphism group size.
• Even when the spread has a 1-dimensional automorphism group, uncountably many non-isomorphic oriented parallelisms can be generated by applying different SO₃ℝ-conjugates.
• The isomorphism type of the parallelism depends not only on the isomorphism type of the spread but also on the specific SO₃ℝ-action used, leading to non-isomorphic structures from the same spread.
• The construction yields more oriented parallelisms with SO₃ℝ symmetry than ordinary parallelisms, due to the richer structure of SO₃ℝ-conjugacy classes in the affine group.
• The Clifford parallelism is the only non-classical parallelism with automorphism group of dimension ≥4, and SO₃ℝ is the largest possible automorphism group for non-Clifford topological parallelisms.