On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

Abstract

ABSTRACT. This paper concerns 4-dimensional (topological locally compact connected) Minkowski planes that admit a 7-dimensional automorphism group E. It is shown that such a plane is either classical or has a distinguished point that is fixed by the connected component of E and that the derived affine plane at this point is a 4-dimensional translation plane with a 7-dimensional collineation group.A Minkowski plane J//= (P, ~, { [I +, II -}) consists of a set of points P, a set of at least two circles ~ (considered as subsets of P) and two equivalence relations I[ + and I[ -on P (parallelisms) such that three pairwise non-parallel points (that is, neither (+)-parallel nor (-)-parallel) can be joined by a unique circle, such that the circles which touch a fixed circle K at p E K partition P\lPl (here [p[ = I pt + u [pl-denotes the union of the two parallel classes of p), such that each parallel class meets each circle in a unique point (parallel projection), such that each (+)-parallel class and each (-)-parallel class intersect in a unique point, and such that there is a circle that contains at least three points (compare [15]). A topological Minkowski plane is a Minkowski plane in which the point set P and the set of circles j~r carry topologies such that the geometric operations of joining, touching, the parallel projections, intersecting parallel classes of different type, and intersecting circles are continuous operations on their domains of definition (see 1-15]). A topological Minkowski plane is called (locally) compact, connected, or finite-dimensional if the point space has the respective topological property. For brevity, a locally compact connected finite-dimensional topological Minkowski plane will be called a finite-dimensional Minkowski plane. According to 1-9, 2.3] a finite-dimensional Minkowski plane can only be of dimension 2 or 4. In these cases the automorphism group of J/is a Lie group with respect to the compact-open topology of dimension at most 6 and 12, respectively, see [16]. The classical model of a 2-or 4-dimensional Minkowski plane is obtained as the geometry of non-trivial plane sections of a ruled quadric in the real or complex projective 3-dimensional space respectively. In these cases the topologies on the point set and the set of circles are induced from the surrounding projective 3-space (the set of planes in the projective 3-space carries a natural topology which can be obtained by duality from the topology of the point set in the 3-space).

On the continuity of the geometric operations in Minkowski planes with parallel classesS 2

“…Furthermore, it was shown in [14] that a 4-dimensional Minkowski plane that admits an 8-dimensional automorphism group must be classical. However, using the results of (15], candidates for 4-dimensional Minkowski planes with a 7-dimensional automorphism groups can be constructed, but it is difficult to verify the axioms of a Minkowski plane. After that one still has to verify the continuity of the geometric operations.…”