A family of 2-dimensional Minkowski planes with small automorphism groups
A topological Minkowski plane is a Minkowski plane whose point set and whose set of circles are endowed with a topology that renders the geometric operations (joining, touching, parallel projection) continuous. If the point set is compact, connected and of finite topological dimension, then the point space is in fact a manifold of dimension 2 or 4 by a result of R. Löwen. Topological Minkowski planes whose point space is a \(d\)-dimensional manifold are called \(d\)-dimensional Minkowski planes. The paper under review concentrates on the 2-dimensional case. The classical representative of this class is the real Minkowski plane, i.e., the geometry of nontrivial plane sections of a ruled quadric in the 3-dimensional real projective space. This classical plane is special with respect to many incidence properties (e.g., Miquel's theorem holds). It is also characterized by the sheer size of its automorphism group, in the following sense. For every 2-dimensional Minkowski plane \(\mathcal M\), the group \(\text{Aut}({\mathcal M})\) of automorphisms of \(\mathcal M\) can be endowed with the compact open topology. This topology renders \(\text{Aut}({\mathcal M})\) a Lie group. Thus one can measure the size of \(\text{Aut}({\mathcal M})\) by its dimension. The real Minkowski plane has an automorphism group of dimension 6. Now a result by A. Schenkel says that a 2-dimensional Minkowski plane is isomorphic to the real Minkowski plane if its automorphism group has dimension at least 5. Schenkel also determined all 2-dimensional Minkowski planes with 4-dimensional automorphism group. The paper under review describes a very general construction method for 2-dimensional Minkowski planes. The isomorphisms between these planes are determined; and the automorphism groups are studied in detail. It turns out that the construction yields planes with automorphism groups of arbitrary dimension \(d \in \{0,1,2,3,4,6\}\). Among the planes with 0- dimensional group, there is also a family of rigid planes; i.e., planes with trivial automorphism group. It becomes obvious that 2-dimensional Minkowski planes with small automorphism groups exist in abundance. (In contrast, no 4-dimensional Minkowski plane is known except the complex Minkowski plane.) Recently, A. E. Schoth described methods to obtain 2-dimensional general quadrangles from 2-dimensional Minkowski planes (more generally, from 2- dimensional circle planes), and vice versa. Thus the construction in the paper under review also yields large classes of 2-dimensional generalized quadrangles, with comparatively small groups of automorphisms.
scientific article; zbMATH DE number 687814