Abstract
The paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E6(−26). A 16-dimensional, compact projective plane P admitting an automorphism group of dimension 41 or more is classical, [18] 87.5 and 87.7. For the special case of a semisimple group Δ acting on P the same result can be obtained if dim δ ≧ 37, see [16]. Our aim is to lower this bound. We show: if Δ is semisimple and dim δ ≧ 29, then P is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin9 (ℝ, r), r ∈ {0, 1 }. The underlying paper contains the first part of the proof showing that Δ is in fact almost simple.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Bödi, On the dimension of automorphism groups of four-dimensional double loops. Math. Z. 215, 89–97 (1994).
R. Bödi, On the dimension of automorphism groups of 8-dimensional ternary fields I. J. Geom. 52, 30–40 (1995).
S. Helgason, Differential geometry, Lie groups, and symmetric spaces. New York 1978.
R. Löwen, Topology and dimension of stable planes: On a conjecture of H. Freudenthal. J. Reine Angew. Math. 343, 180–122 (1983).
D. Montgomery and H. Samelson, Transformation groups of spheres. Ann. of Math. 44, 454–470 (1943).
B. Priwitzer, Kompakte 16-dimensionale Ebenen mit großen halbeinfachen Gruppen. Thesis, Tübingen 1994.
B. Priwitzer, Large automorphism groups of 8-dimensional projective planes are Lie groups. Geom. Dedicata 52, 33–40 (1995).
B. Priwitzer, Compact Groups on Compact Projective Planes. Geom. Dedicata 58, 245–258 (1995).
B. Priwitzer, Almost simple groups acting on 16-dimensional projective planes. Preprint.
B. Priwitzer and H. Salzmann, Large automorphism groups of 16-dimensional planes are Lie groups. Preprint.
R. W. Richardson, Groups acting on the 4-sphere. Illinois J. Math. 5, 474–485 (1961).
H. Salzmann, Compact 8-dimensional projective planes with large collineation groups. Geom. Dedicata 8, 139–161 (1979).
H. Salzmann, Automorphismengruppen 8-dimensionaler Ternärkörper. Math. Z. 166, 265–275 (1979).
H. Salzmann, Kompakte, 8-dimensionale projektive Ebenen mit großer Kollineationsgruppe. Math. Z. 176, 345–357 (1981).
H. Salzmann, Baer-Kollineationsgruppen der klassischen projektiven Ebenen. Arch. Math. 38, 374–377 (1982).
H. Salzmann, Compact 16-dimensional projective planes with large collineation groups. Math. Ann. 261, 447–454 (1982).
H. Salzmann, Compact 16-dimensional projective planes with large collineation groups. II. Monatsh. Math. 95, 311–319 (1983).
H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen and M. Stroppel, Compact projective planes. Berlin 1995.
J. Tits, Sur certaines classes d’espaces homogènes de groupes die Lie. Mém. Acad. Belg. 8° 29, 1–268 (1955).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Priwitzer, B. Large semisimple groups on 16-dimensional compact projective planes are almost simple. Arch. Math. 68, 430–440 (1997). https://doi.org/10.1007/s000130050075
Received:
Issue Date:
DOI: https://doi.org/10.1007/s000130050075