4-dimensional compact projective planes with a 5-dimensional nilradical
type: Item
instance of: scholarly article
title: 4-dimensional compact projective planes with a 5-dimensional nilradical
zbMATH Open document ID: 0843.51018
DOI: 10.1007/BF01263457
author: Dieter Betten
published in: Geometriae Dedicata
publication date: 1996-08-18T00:00:00Z
review text: The automorphism group \(\Sigma\) of a 4-dimensional compact projective plane \(\mathcal P\) is at most 16-dimensional, and if \(\dim \Sigma \geq 9\), then \(\mathcal P\) is the classical complex plane and \(\Sigma \cong \text{PSL}_3 \triangleleft\), [the reviewer et al., Compact Projective Planes, Berlin (1995), 73.22]. All planes \(\mathcal P\) with \(\dim \Sigma = 8\) or 7 have been determined explicitly. If \(\dim \Sigma = 8\), then \(\mathcal P\) is a translation plane, [loc. cit.]. For \(\dim \Sigma = 7\) the plane \(\mathcal P\) is either isomorphic to a (dual) translation plane or to a so-called shift plane [the author, Geom. Dedicata 36, No. 2/3, 151-170 (1990; Zbl 0717.51013)]. The present paper deals with the case \(\dim \Sigma = 6\). In addition, the author assumes that \(\Sigma\) has an open orbit in the space of flags, i.e., that \(\mathcal P\) is flexible. If \(\Sigma\) is not solvable, then its connected component \(\Delta\) is isomorphic \(\gtreqless^2 \cdot GL_2 \gtreqless\), [\textit{R. Löwen}, Geom. Dedicata 36, No. 2/3, 225-234 (1990; Zbl 0712.51011)]. Planes with such a group are semitranslation planes as will be proved in a forthcoming paper by H. Klein, N. Knarr, and R. Löwen. Thus, one may further assume that \(\Delta\) is solvable and it can be shown that \(\Delta\) fixes a flag \((\infty, W)\). Now several cases can be distinguished according to the dimension of the nil radical \(\mathcal N\) of \(\Delta\). \(\dim {\mathcal N} = 6\) leads to a single shift plane [the author, Mitt. Math. Ges. Hamb. 12, No. 3, 741-747 (1991; Zbl 0749.51015)]. If \(\dim {\mathcal N} = 5\) one proceeds by considering the orbits of \(\Delta\) on the line \(W\) and on the line pencil through \(\infty\), [the author, J. Geom. 42, No. 1/2, 30-40 (1991; Zbl 0745.51009)]. The possible four cases for such orbits are treated separately in this paper and as a result the author obtains four new families of 4-dimensional compact projective planes which are neither transl…
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 51H20
Mathematics Subject Classification ID: 51A35
zbMATH DE Number: 826786
zbMATH Keywords: topological planes
zbMATH Keywords: solvable Lie groups
zbMATH Keywords: flexible 4-dimensional planes
reviewed by: Helmut Salzmann
MaRDI profile type: Publication
cites work: 4-dimensionale projektive Ebenen mit 3-dimensionaler Translationsgruppe
cites work: 4-dimensional compact projective planes with a 7-dimensional collineation group
cites work: Q4000946
cites work: Orbits in 4-dimensional compact projective planes
cites work: Sur Les Représentations Unitaires Des Groupes De Lie Nilpotents. III
cites work: Covering spaces, fibre spaces, and local homeomorphisms
cites work: Homologies and elations in compact, connected projective planes
cites work: Q3788625
cites work: Ein Kriterium für topologische Ternärkörper
cites work: 4-dimensionale projektive Ebenen vom Lenz-Barlotti-Typ II.2. (Four- dimensional projective planes of Lenz-Barlotti type II.2.)
cites work: 4-dimensionale projektive Ebenen mit großer abelscher Kollineationsgruppe. (Four-dimensional projective planes with large abelian collineation group)
cites work: Four-dimensional compact projective planes with a nonsolvable automorphism group
cites work: Q5652137
cites work: Topological planes
cites work: Kollineationsgruppen kompakter, vier-dimensionaler Ebenen
cites work: Kollineationsgruppen kompakter 4-dimensionaler Ebenen. II
cites work: Elations in four-dimensional planes
Recommended article: Orbits in 4-dimensional compact projective planes
Recommended article: Q4358762
Recommended article: Four-dimensional compact projective planes with a solvable collineation group
Recommended article: 4-dimensional compact projective planes with a 7-dimensional collineation group
Recommended article: Q3788625
Recommended article: Q3761207
Recommended article: Four-dimensional compact projective planes with collineation group \(N_{6,28}\)
Recommended article: 4-dimensional compact projective planes with small nilradical
Recommended article: Q1090936
Recommended article: Q4408476
description: scientific article; zbMATH DE number 826786