The piecewise projective group as group of projectivities
A flat Minkowski plane consists of a system of circles which are graphs of homeomorphisms of the 1-sphere \(\mathbb{S}^1\) on the torus \(\mathbb{S}^1\times\mathbb{S}^1\) such that three mutually non-parallel points can uniquely be joined by a circle and such that the axiom of touching is satisfied. The set of circles is a locally compact topological space with two connected components, one consisting of all the graphs of orientation preserving homeomorphisms the other of all the graphs of orientation reversing homeomorphisms. Given the same basic representation of two Minkowski planes one can exchange the same kind of connected component and obtain again flat Minkowski planes as was shown by the author and the reviewer [Can. Math. Bull. 38, 469-480 (1995; Zbl 0844.51005)]. The paper under review contains a new construction of flat Minkowski planes from two given flat Minkowski planes which refines the above mentioned method. In this construction one considers a flat Minkowski plane relative to a point \(p\). Circles in the component that are made up of graphs of orientation preserving homeomorphisms are of two kinds. These are the circles that pass through \(p\) and the other circles. Assuming that all translations of the derived affine plane at the distinguished point \(p\) are automorphisms of the Minkowski plane, one obtains relative to this derived affine plane a picture of straight lines and certain curves that look like hyperbolas. Each of these curves has two branches, one convex and the other concave. The convex branches taken together form, what the author calls a quarter of the Minkowski plane, and similarly for the concave branches. The author shows that the convex branches in one quarter of such a Minkowski plane and the concave branches in a quarter of another Minkowski plane plus the lines that come from graphs of orientation preserving homeomorphisms can be put together via a pasting function, i.e., an increasing homeomorphism of \(\mathbb{R}^+\), to form half the circle set of a flat Minkowski plane that again admits all translations as automorphisms. The pasting function determines which branches are joined to form a circle in one pencil; the other circles are obtained by translation. Then, by adding a suitable half of a Minkowski plane, one finally obtains a flat Minkowski plane. Many of the known families of flat Minkowski planes can be obtained as special cases of this construction. Besides this, many new planes can be formed. The author modifies his construction in a forthcoming paper to obtain proper flat hyperbola planes or toroidal planes, that is, planes that look like flat Minkowski planes except that the axiom of touching is not satisfied. In the second part of the paper the groups of projectivities of such planes are investigated. In general, this group becomes larger as the homogeneity, i.e., the dimension of the automorphism group of the plane decreases. For special choices of the pasting functions and starting with the classical Minkowski plane the author determines the groups of projectivities of the corresponding planes. Using characterizations obtained by \textit{D. Betten} [J. Geom. 13, 197-209 (1919; Zbl 0426.51009)] in the determination of the groups of projectivities of Moulton planes, the author shows that in these cases the group of piecewise projective transformations of the projective line results.
scientific article; zbMATH DE number 933261