Parabolic and unipotent collineation groups of locally compact connected translation planes

Abstract

A closed connected subgroup $\mathrm{\Gamma }$ of the reduced stabilizer $S{\mathbb{G}}_0$ of a locally compact connected translation plane $\left(P,\mathcal{ℒ}\right)$ is called parabolic, if it fixes precisely one line $S\in {\mathcal{ℒ}}_0$ and if it contains at least one compression subgroup. We prove that $\mathrm{\Gamma }$ is a semidirect product of a stabilizer ${\mathrm{\Gamma }}_W$, where $W$ is a line of ${\mathcal{ℒ}}_0\setminus \left\{S\right\}$ such that ${\mathrm{\Gamma }}_W$ contains a compression subgroup, and the commutator subgroup $R^{\prime }$ of the radical $R$ of $\mathrm{\Gamma }$. The stabilizer ${\mathrm{\Gamma }}_W$ is a direct product ${\mathrm{\Gamma }}_W=K\times \Upsilon$ of a maximal compact subgroup $K\le \mathrm{\Gamma }$ and a compression subgroup $\Upsilon$. Therefore, we have a decomposition $\mathrm{\Gamma }=K\cdot \Upsilon \cdot N$ similar to the Iwasawa-decomposition of a reductive Lie group.

Such a “geometric Iwasawa-decomposition” $\mathrm{\Gamma }=K\cdot \Upsilon \cdot N$ is possible whenever $\mathrm{\Gamma }\le S{\mathbb{G}}_0$ is a closed connected subgroup which contains at least one compression subgroup $\Upsilon$. Then the set $\mathcal{S}$ of all lines through $0$ which are fixed by some compression subgroup of $\mathrm{\Gamma }$ is homeomorphic to a sphere of dimension $dimN$. Removing the $\mathrm{\Gamma }$-invariant lines from $\mathcal{S}$ yields an orbit of $\mathrm{\Gamma }$.

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 $N=N_1\times \mathrm{\Sigma }$ of a central subgroup $\mathrm{\Sigma }$ consisting of all shears in $N$ and a complementary normal subgroup $N_1$ which contains the commutator subgroup $N^{\prime }$ of $N$.