Benz planes of Dembowski type and groups of tangential projectivities
A Benz plane \(M\) is said to be of Dembowski type if the tangency relation of \(M\) behaves like the tangency relation in a finite Benz plane of even order. For a Minkowski, respectively Möbius plane, this means that for each bundle \(\mathcal B\) of circles there exists a bundle, respectively a flock \(\mathcal B'\), such that each circle of \(\mathcal B\) is tangent to each circle of \(\mathcal B'\). For a Laguerre plane this means that for every tangent pencil \(\mathcal B\) and every point \(r\) different from the carrier of \(\mathcal B\) there exists a tangent pencil \(\mathcal B'\) with carrier \(r\) such that each circle of \(\mathcal B'\) is tangent to each circle of \(\mathcal B\). Using the tangency relation, in Benz planes of Dembowski type one can define tangential perspectivities and projectivities. The authors show that a Benz plane \(M\) of Dembowski type is Miquelian if and only if every tangential perspectivity is induced by an automorphism of \(M\). There are also several results on the group of tangential projectivities in non-Miquelian Benz planes of Dembowski type, especially in the finite case.
scientific article; zbMATH DE number 6023282
scientific article