Abstract
We consider a four-dimensional compact projective plane π=(\(\mathcal{P}\), \(\mathcal{L}\)) whose collineation group Σ is six-dimensional and solvable with a nilradical N isomorphic to Nil × R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that Σ fixes a flag p∈W, acts transitively on \(\mathcal{L}\) p \∖{W}, and fixes no point in the set W∖{p}. We study the actions of Σ and N on \(\mathcal{P}\) and on the pencil \(\mathcal{L}\) p \∖{W}, in the case that Σ does not contain a three-dimensional elation group. In the special situation that Σ acts doubly transitively on \(\mathcal{L}\) p ∖{W}, we will determine all possible planes π. There are exactly two series of such planes.
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Klein, H. Four-dimensional compact projective planes with doubly transitive action on the fixed pencil. Geom Dedicata 61, 227–255 (1996). https://doi.org/10.1007/BF00150025
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DOI: https://doi.org/10.1007/BF00150025