Four-dimensional compact projective planes with two fixed points (Q1911996)

The paper under review studies topological projective planes whose point space is a four-dimensional topological manifold. Endowed with the compact-open topology, the connected component \(\Sigma\) of the group of all continuous collineations of such a plane is a Lie group acting continuously on the space of points, lines and flags. Of particular interest is the case where \(\Sigma\) has an open orbit in the space of flags, which allows to reconstruct the plane from the given action of the group. In this case, one has \(\dim \Sigma \geq 6\). The authors assume in addition that \(\Sigma\) is solvable, fixes two points, and acts transitively on the complement of the joining line in both of the corresponding line pencils. Using the classification of solvable Lie algebras of low dimension (via their nilpotent radical), the authors prove that these assumptions characterize a family of planes which can be described by a single construction, involving two real parameters. Except for a single pair of parameters, these planes appear to be new examples of four-dimensional compact projective planes.