Innovations in Incidence Geometry

Parabolic and unipotent collineation groups of locally compact connected translation planes

Harald Löwe

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A closed connected subgroup Γ of the reduced stabilizer S G 0 of a locally compact connected translation plane ( P , ) is called parabolic, if it fixes precisely one line S 0 and if it contains at least one compression subgroup. We prove that Γ is a semidirect product of a stabilizer Γ W , where W is a line of 0 { S } such that Γ W contains a compression subgroup, and the commutator subgroup R of the radical R of Γ . The stabilizer Γ W is a direct product Γ W = K × ϒ of a maximal compact subgroup K Γ and a compression subgroup ϒ . Therefore, we have a decomposition Γ = K ϒ N similar to the Iwasawa-decomposition of a reductive Lie group.

Such a “geometric Iwasawa-decomposition” Γ = K ϒ N is possible whenever Γ S G 0 is a closed connected subgroup which contains at least one compression subgroup ϒ . Then the set S of all lines through 0 which are fixed by some compression subgroup of Γ is homeomorphic to a sphere of dimension dim N . Removing the Γ -invariant lines from S yields an orbit of Γ .

Furthermore, we consider closed connected subgroups N S G 0 whose Lie algebra consists of nilpotent endomorphisms of P . Our main result states that N is a direct product N = N 1 × Σ of a central subgroup Σ consisting of all shears in N and a complementary normal subgroup N 1 which contains the commutator subgroup N of N .

Article information

Innov. Incidence Geom., Volume 2, Number 1 (2005), 57-82.

Received: 7 October 2004
Accepted: 20 January 2005
First available in Project Euclid: 28 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51A10: Homomorphism, automorphism and dualities 51A49 51H10: Topological linear incidence structures

affine translation plane automorphism group parabolic collineation group hinge group unipotent collineation group shears weight line weight sphere


Löwe, Harald. Parabolic and unipotent collineation groups of locally compact connected translation planes. Innov. Incidence Geom. 2 (2005), no. 1, 57--82. doi:10.2140/iig.2005.2.57.

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