Innovations in Incidence Geometry
- Innov. Incidence Geom.
- Volume 2, Number 1 (2005), 57-82.
Parabolic and unipotent collineation groups of locally compact connected translation planes
A closed connected subgroup of the reduced stabilizer of a locally compact connected translation plane is called parabolic, if it fixes precisely one line and if it contains at least one compression subgroup. We prove that is a semidirect product of a stabilizer , where is a line of such that contains a compression subgroup, and the commutator subgroup of the radical of . The stabilizer is a direct product of a maximal compact subgroup and a compression subgroup . Therefore, we have a decomposition similar to the Iwasawa-decomposition of a reductive Lie group.
Such a “geometric Iwasawa-decomposition” is possible whenever is a closed connected subgroup which contains at least one compression subgroup . Then the set of all lines through which are fixed by some compression subgroup of is homeomorphic to a sphere of dimension . Removing the -invariant lines from yields an orbit of .
Furthermore, we consider closed connected subgroups whose Lie algebra consists of nilpotent endomorphisms of . Our main result states that is a direct product of a central subgroup consisting of all shears in and a complementary normal subgroup which contains the commutator subgroup of .
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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. https://projecteuclid.org/euclid.iig/1551323262