Abstract
We consider 4-dimensional flexible projective planes with the following properties: The collineation group is a 6-dimensional solvable Lie group which fixes some flag ∞ ∈ W. Furthermore, the collineation group has a 1-dimensional orbit both on W and on the pencil of lines through {∞}. We show that there are three different families of planes with these properties.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
D. Betten, 4-dimensionale Translationsebenen mit genau einer Fixrichtung, Geom. Dedicata 3 (1975), 405–440
D. Betten, 4-dimensionale projektive Ebenen mit 3-dimensionaler Translationsgruppe, Geom. Dedicata 16 (1984), 179–193
D. Betten, 4-dimensional compact projective planes with a 7-dimensional collineation group, Geom. Dedicata 36 (1990), 151–170
D. Betten, 4-dimensional compact projective planes with a nilpotent collineation group, Mitt. Math. Ges. Hamburg 12 (1991), 741–747
D. Betten, Orbits in 4-dimensional compact projective planes, J. of Geometry 42 (1991), 30–40
D. Betten and H. Klein, 4-dimensional compact projective planes with two fixed points, J. of Geometry 55 (1996), 31–56
D. Betten, 4-dimensional compact projective planes with a 5-dimensional nilradical, Geom. Dedicata 58 (1995), 259–289
H. Hähl, Homologies and dations in compact, connected projective planes, Topology Appl. 12 (1981), 49–63
N. Knarr, Topologische Differenzenflächenebenen, Diplomarbeit Kiel, 1983
N. Knarr, 4-dimensionale projektive Ebenen vom Lenz-Barlotti-Typ II.2, Results in Math. 12 (1987), 114–124
R. Löwen, Four-dimensional compact projective planes with a nonsolvable automorphism group, Geom. Dedicata 36 (1990), 225–234
G. M. Mubarakzjanov, Classification of solvable Lie algebras of sixth order with one non-nilpotent basis element, Izv. Vyssh. Uchebn. Zaved. Matematica 4 (1963), 104–116
H. R. Salzmann, Kollineationsgruppen kompakter 4-dimensionaler Ebenen, Math. Z. 117 (1970), 112–124
H. R. Salzmann, Kollineationsgruppen kompakter 4-dimensionaler Ebenen II, Math. Z. 121 (1971), 104–110
H. R. Salzmann — D. Betten — T. Grundhöfer — H. Hähl — R. Löwen — M. Stroppel, Compact Projective Planes, de Gruyter Berlin, 1995
P. Turkowski, Solvable Lie algebras of dimension six, J. Math. Phys. 31 (1990), 1344–1350
C. Weigand, Konstruktion topologischer projektiver Ebenen, die keine Translationsebenen sind, Mitt. Math. Sem. Giessen 177 (1987)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Betten, D., Polster, B. Four-Dimensional Compact Projective Planes of Orbit Type (1,1). Results. Math. 36, 208–236 (1999). https://doi.org/10.1007/BF03322112
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322112