Affine Hughes groups acting on 4-dimensional compact projective planes
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Rainer Löwen
Abstract
Suppose that the real projective plane is contained as a Baer subplane 
in a compact 4-dimensional plane 
. Salzmann [15] has shown that if the group of projective collineations of extends to , then is isomorphic to the complex plane. This means that there are no compact 4-dimensional Hughes planes. If the group AGL2ℝ of offine collineations of extends to , then we call the extended group an offine Hughes group. Here, non-classical examples exist. We show that, up to duality, every action of AGL2ℝ on a compact 4-dimensional plane is an offine Hughes group, and we describe explicitly the possible actions on the point space of as transformation groups. In subsequent work (jointly with N. Knarr and H. Klein), this will be used to determine the possible planes , as well. This will complete the classification of 4-dimensional planes admitting a non-solvable group of dimension at least 6.
© Walter de Gruyter
Articles in the same Issue
- Boundary value problem with an oblique derivative for uniformly elliptic operators with discontinuous coefficients
- Radon transforms on the symmetric group and harmonic analysis of a class of invariant Laplacians
- Linear submanifolds and bisectors in ℂHn
- Affine Hughes groups acting on 4-dimensional compact projective planes
- A parallelism for contact conformal sub-Riemannian geometry
- Finite groups with smooth one fixed point actions on spheres
Articles in the same Issue
- Boundary value problem with an oblique derivative for uniformly elliptic operators with discontinuous coefficients
- Radon transforms on the symmetric group and harmonic analysis of a class of invariant Laplacians
- Linear submanifolds and bisectors in ℂHn
- Affine Hughes groups acting on 4-dimensional compact projective planes
- A parallelism for contact conformal sub-Riemannian geometry
- Finite groups with smooth one fixed point actions on spheres
