On non-projective free Benz planes (Q1808800)

Free extensions are often used in geometry to show the existence of certain models and to construct examples whose properties are contrary to those of common models. Many of the methods used on the individual planes are similar but quite distinct. In [Rend. Semin. Mat. Brescia 7, 399-407 (1984; Zbl 0543.51005)] \textit{O. Iden} gave a unified treatment for the classification of free projective Benz planes, that is, Möbius, Laguerre and Minkowski planes in which every two circles have (at least) one point in common. In the paper under review the authors present a revision of Iden's paper for non-projective free Benz planes (here there are circles that have no point in common) and give a unified treatment of such planes. Equivalence between such planes is based on the concept of simple extensions in locally affine planes, that is, affine, projective, and Benz planes. One essentially has a partial plane generated naturally from a given partial plane by an additional single element under a minimal construction process. Two partial planes are freely equivalent if one can be obtained from the other by a sequence of simple extensions. Using the extracidence graph associated with a locally affine plane and \(\alpha\)-reductions the authors classify open, finite, non-degenerate, non-projective, partial Benz planes under free equivalence. They show that such a plane is characterised by its rank except when, in the Laguerre case, it is freely equivalent to a configuration discovered by T. E. Guleng. The authors also include a new proof of the well known fact that for any given group \(G\) there exists a Möbius plane that has \(G\) as its full automorphism group. It uses a characterisation of free planar extensions of normal partial Möbius planes and ideas of \textit{E. Mendelsohn} [J. Geom. 2, 97-106 (1972; Zbl 0242.50013)] and is based on similar theorems concerning projective planes and a particular configuration of S. Ditor.