#### 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 $\Gamma$ of the reduced stabilizer $S{\mathbb{G}}_{0}$ of a locally compact connected translation plane $\left(P,\mathsc{ℒ}\right)$ is called parabolic, if it fixes precisely one line $S\in {\mathsc{ℒ}}_{0}$ and if it contains at least one compression subgroup. We prove that $\Gamma$ is a semidirect product of a stabilizer ${\Gamma }_{W}$, where $W$ is a line of ${\mathsc{ℒ}}_{0}\setminus \left\{S\right\}$ such that ${\Gamma }_{W}$ contains a compression subgroup, and the commutator subgroup ${R}^{\prime }$ of the radical $R$ of $\Gamma$. The stabilizer ${\Gamma }_{W}$ is a direct product ${\Gamma }_{W}=K×\Upsilon$ of a maximal compact subgroup $K\le \Gamma$ and a compression subgroup $\Upsilon$. Therefore, we have a decomposition $\Gamma =K\cdot \Upsilon \cdot N$ similar to the Iwasawa-decomposition of a reductive Lie group.

Such a “geometric Iwasawa-decomposition” $\Gamma =K\cdot \Upsilon \cdot N$ is possible whenever $\Gamma \le S{\mathbb{G}}_{0}$ is a closed connected subgroup which contains at least one compression subgroup $\Upsilon$. Then the set $\mathsc{S}$ of all lines through $0$ which are fixed by some compression subgroup of $\Gamma$ is homeomorphic to a sphere of dimension $dimN$. Removing the $\Gamma$-invariant lines from $\mathsc{S}$ yields an orbit of $\Gamma$.

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