Four-dimensional compact projective planes admitting an affine Hughes group (Q1840588)

The authors determine all 4-dimensional affine Hughes planes that are not translation planes. Thereby they finalize the classification of all compact 4-dimensional projective planes \({\mathbf P}\) having a non-solvable automorphism group of dimension at least 6 [cf. \textit{R. Löwen}, Geom. Dedicata 36, 225-234 (1990; Zbl 0712.51011)]. In the solvable case a classification has been given by \textit{H. Klein} [Geom. Dedicata 77, 271-277 (1999; Zbl 0951.51006)]. The authors use \textit{R. Löwen}'s results [Forum Math. 10, 435-451, (1998; Zbl 0914.51014)] on \({\mathbf P}\), saying that up to duality such planes (apart from one exceptional plane) admit an affine Hughes group isomorphic to the group \(\Delta\) of orientation preserving affine maps of \(\mathbb{R}^2\). \(\Delta\) fixes a line \(W\), and Löwen determined all possible actions of \(\Delta\) on the point set \(P\setminus W\) of the affine plane derived from \({\mathbf P}\). Another tool is \textit{N. Knarr}'s construction principle from [Bull. Belgian Math. Soc. Simon Stevin 7, 61-71 (2000; Zbl 0952.51003)]. Knarr described affine planes \(A\) on \(\mathbb{R}^2\times \mathbb{R}^2\) (admitting the classical action of \(\mathbb{R}^2\cdot SL_2\mathbb{R} <\Delta)\) in terms of two continuous functions \(f_1,f_2:\mathbb{R}^2 \to\mathbb{R}^2\) defining the line set of \(A\). In the paper, it is shown (3) how to choose \(f_1,f_2\) according to the type of action considered from Löwen's list. The main part of the work (4-10) is devoted to proving Knarr's conditions on \(f_1,f_2\). This includes solving certain nonlinear systems of two equations in two variables (which encode the affine structure) using topological degree techniques and sophisticated calculations. Finally (11-13), continuity of the geometric operations is proved.