Compactness of the automorphism group of a topological parallelism on real projective 3-space: The disconnected case

Löwen, Rainer

doi:10.36045/bbms/1546570914

Cited by 7 publications

(15 citation statements)

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“…This happens, for instance, if H is chosen to be the skew field of Hamilton's quaternions over the real numbers. (It is worth noting that D. Betten, R. Löwen and R. Riesinger characterised Clifford parallelism among the topological parallelisms of the 3dimensional real projective space by its (linear) automorphism group in [2], [4], [25], [26], [27].) The next step is to consider the (full) automorphism group Γ .…”

Section: Introductionmentioning

confidence: 99%

Clifford-like parallelisms

2018

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In this paper we focus on the description of the automorphism group Γ of a Clifford-like parallelism on a 3-dimensional projective double space P(H F ), ℓ , r over a quaternion skew field H (with centre a field F of any characteristic). We compare Γ with the automorphism group Γ ℓ of the left parallelism ℓ , which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose in such a way that Γ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ℓ .Mathematics Subject Classification (2010): 51A15, 51J15

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Section: Introductionmentioning

confidence: 99%

Clifford-like parallelisms

2018

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“…The properties of the two kinds of parallelisms are very similar. See, e.g., the section 'Preliminaries' of [13] for a brief account of the essentails in the ordinary case. One distinctive feature is the homeomorphism type of a parallelism, considered as a subset of the hyperspace of L or L + : PROPOSITION 2.3 An ordinary topological parallelism Π is homeomorphic to the real projective plane.…”

Section: Parallelisms Versus Oriented Parallelismsmentioning

confidence: 99%

“…The same proof goes through in the oriented case virtually unchanged. Lemma 3.4 of [13] states that every automorphism σ induces equivalent actions on the line stars of any two fixed points. Of course, this has to be adapted by using the oriented line stars.…”

Section: Automorphismsmentioning

confidence: 99%

“…Proof. We know that Σ is compact [13] and contains the 3-dimensional group Ω, and Σ is at most 3-dimensional because Π is not Clifford by assumption [12], hence Σ 1 = Ω. Now Σ is contained in a maximal compact subgroup PO(4, R) = Z 2 ·PSO 4 R of PGL(4, R).…”

Section: Proof Assertionmentioning

confidence: 99%

“…This proves that Σ = Ω. As an alternative argument, one could use Lemma 3.4 of [13] in order to rule out the possibility ̺ ∈ Σ. PROPOSITION 5.9 Let two SO 2 R-rotational spreads C, C ′ and Γ-admissible copies Ω and Ω ′ of SO 3 R be given.…”

Section: Proof Assertionmentioning

confidence: 99%

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Rotational spreads and rotational parallelisms and oriented parallelisms of $${\mathrm{PG}(3,{\mathbb {R}})}$$ PG ( 3 , R )

Löwen

2019

J. Geom.

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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 R-action (only the Clifford parallelism admits a larger group [12]), and it turns out surprisingly that there are far more oriented parallelisms of this kind than ordinary parallelisms.More specifically, Betten and Riesinger [5] construct ordinary parallelisms by applying SO 3 R to rotational Betten spreads. We show that these are the only ordinary parallelisms compatible with this group action, but also the 'acentric' rotational spreads considered by them yield oriented parallelisms. The automorphism group of the resulting (oriented or non-oriented) parallelisms is always SO 3 R, no matter how large the automorphism group of the non-regular spread is. The isomorphism type of the parallelism depends not only on the isomorphism type of the spread used, but also on the rotation group applied to it. We also study the rotational Betten spreads used in this construction and their automorphisms.MSC 2010: 51H10, 51A15, 51M30

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Parallelisms of $$\mathrm{PG}(3,\mathbb R)$$ PG ( 3 , R ) admitting a 3-dimensional group

Löwen

2018

Beitr Algebra Geom

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Betten and Riesinger [1] constructed Parallelisms of PG(3, R) with automorphism group SO 3 R by applying the reducible SO 3 R-action to a rotational Betten spread. This was generalized in [8] so as to include oriented parallelisms (i.e., parallelisms of oriented lines). In this way, a much larger class of examples was produced.Here we show that, apart from Clifford parallelism, these are the only topological parallelisms admitting an automorphism group of dimension 3 or larger. In particular, we show that a topological parallelism admitting the irreducible action of SO 3 R is Clifford.MSC 2010: 51H10, 51A15, 51M30

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Compactness of the automorphism group of a topological parallelism on real projective 3-space: The disconnected case

Abstract: We prove that the automorphism group of a topological parallelism on real projective 3-space is compact. This settles a conjecture stated in [1], where it was proved that at least the connected component of the identity is compact.MSC 2000: 51H10, 51A15, 51M30

Cited by 7 publications

References 8 publications

Clifford-like parallelisms

Clifford-like parallelisms

Rotational spreads and rotational parallelisms and oriented parallelisms of $${\mathrm{PG}(3,{\mathbb {R}})}$$ PG ( 3 , R )

Parallelisms of $$\mathrm{PG}(3,\mathbb R)$$ PG ( 3 , R ) admitting a 3-dimensional group

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