Compactness of the Automorphism Group of a Topological Parallelism on Real Projective 3-Space

Betten, Dieter; Löwen, Rainer

doi:10.1007/s00025-017-0674-8

Cited by 10 publications

(27 citation statements)

References 8 publications

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“…The coefficients are polynomials in n whose degree increases as we move away from the main diagonal of the matrix. From this observation it follows that the dynamics of γ acting on P and on L is the same as for a nontrivial cyclic subgroup of a one-parameter group of type (c1) as considered in [1]. This means that p γ ±n converges to p 1 = e 1 as n → ∞, for points p / ∈ H = e ⊥ 4 .…”

Section: R Löwenmentioning

confidence: 75%

“…The proof for the connected case given in [1] rests on the theorem of Malcev-Iwasawa, which implies that a non-compact Lie group contains some closed but non-compact oneparameter subgroup. This reduced the proof to the examination of all possible oneparameter groups.…”

Section: Foundations Of the Proofmentioning

confidence: 99%

“…We shall describe the necessary changes and in one case we shall give an alternative proof. Some simplification over the proof in [1] is achieved by shifting subcases from the 'c' set to the 'a' or 'b' sets. Proof.…”

Section: R Löwenmentioning

confidence: 99%

“…This is the most subtle part of the whole proof. Therefore, we briefly give the argument, although it is almost identical to the one used in [1], 3.5. The matrix γ can be written as 1 + R, where 1 denotes the unit matrix and…”

Section: R Löwenmentioning

confidence: 99%

“…Type (c5). Both proofs of [1], 3.13 can be used without change. Note that we need only consider the case where the eigenvalues are pairwise different.…”

Section: R Löwenmentioning

confidence: 99%

See 4 more Smart Citations

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

Löwen¹

2018

Bull. Belg. Math. Soc. Simon Stevin

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Section: R Löwenmentioning

confidence: 75%

Section: Foundations Of the Proofmentioning

confidence: 99%

Section: R Löwenmentioning

confidence: 99%

Section: R Löwenmentioning

confidence: 99%

“…Type (c5). Both proofs of [1], 3.13 can be used without change. Note that we need only consider the case where the eigenvalues are pairwise different.…”

Section: R Löwenmentioning

confidence: 99%

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

Löwen¹

2018

Bull. Belg. Math. Soc. Simon Stevin

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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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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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Compactness of the Automorphism Group of a Topological Parallelism on Real Projective 3-Space

Abstract: We conjecture that the automorphism group of a topological parallelism on real projective 3-space is compact. We prove that at least the identity component of this group is, indeed, compact.MSC 2000: 51H10, 51A15, 51M30

Cited by 10 publications

References 8 publications

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

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

Clifford-like parallelisms

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

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