## Innovations in Incidence Geometry

### Parabolic and unipotent collineation groups of locally compact connected translation planes

Harald Löwe

#### Abstract

A closed connected subgroup $Γ$ of the reduced stabilizer $S G 0$ of a locally compact connected translation plane $( P , ℒ )$ is called parabolic, if it fixes precisely one line $S ∈ ℒ 0$ 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 $ℒ 0 ∖ { S }$ 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

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

Dates
Accepted: 20 January 2005
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323262

Digital Object Identifier
doi:10.2140/iig.2005.2.57

Mathematical Reviews number (MathSciNet)
MR2214714

Zentralblatt MATH identifier
1182.51004

#### Citation

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

#### References

• N. Bourbaki, Groupes et algèbres de Lie, Chapitres 7 et 8. Hermann (1975).
• T. Buchanan and H. Hähl, On the kernel and the nuclei of $8$-dimensional locally compact quasifields, Arch. Math. 29 (1977), 472–480.
• H. Hähl, Lokalkompakte zusammenhängende Translationsebenen mit großen Sphärenbahnen auf der Translationsachse, Result. Math. 2 (1979), 62–87.
• J. Dixmier, L'application exponentielle dans les groupes de Lie résolubles, Bull. Soc. Math. France 85 (1957), 113–121.
• J. Hilgert and K.-H. Neeb, Lie–Gruppen und Lie–Algebren. Vieweg (1991).
• K. H. Hofmann and A. Mukherjea, On the density of the image of the exponential function, Math. Ann. 234 (1978), 263–273.
• A. W. Knapp, Lie groups beyond an introduction. Birkhäuser (1996).
• H. Löwe, Noncompact, almost simple groups operating on locally compact, connected translation planes, J. Lie Theory 10 (2000), 127–146.
• ––––, A rough classification of symmetric planes, Adv. Geom. 1 (2001), 1–21.
• ––––, Noncompact subgroups of the reduced stabilizer of a locally compact, connected translation plane, Forum Math. 15 (2003), 247–260.
• K.-H. Neeb, Weakly exponential Lie groups, J. Algebra 179 (1996), 331–361.
• H. Salzmann, Topological planes, Adv. Math. 2 (1961), 474–485.
• H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen and M. Stroppel, Compact Projective Planes, Walter de Gruyter (1995).
• V. S. Varadarajan, Lie Groups, Lie Algebra, and Their Representations, Springer (1984).
• M. Wüstner, Contributions to the structure theory of solvable Lie groups. Dissertation (German), Technische Hochschule Darmstadt 1995.