Vol. 2, No. 1, 2005

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Parabolic and unipotent collineation groups of locally compact connected translation planes

Harald Löwe

Vol. 2 (2005), No. 1, 57–82
DOI: 10.2140/iig.2005.2.57
Abstract

A closed connected subgroup Γ of the reduced stabilizer S G0 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 ΓW, where W is a line of 0 {S} such that ΓW contains a compression subgroup, and the commutator subgroup R of the radical R of Γ. The stabilizer ΓW is a direct product ΓW = 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 G0 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 dimN. Removing the Γ-invariant lines from S yields an orbit of Γ.

Furthermore, we consider closed connected subgroups N S G0 whose Lie algebra consists of nilpotent endomorphisms of P. Our main result states that N is a direct product N = N1 × Σ of a central subgroup Σ consisting of all shears in N and a complementary normal subgroup N1 which contains the commutator subgroup N of N.

Keywords
affine translation plane, automorphism group, parabolic collineation group, hinge group, unipotent collineation group, shears, weight line, weight sphere
Mathematical Subject Classification 2000
Primary: 51A10, 51H10
Milestones
Received: 7 October 2004
Accepted: 20 January 2005
Authors
Harald Löwe