Projektivitätengruppen in angeordneten und topologischen Ebenen. (Groups of projectivities in ordered and topological planes)
The influence of the structure of the group \(\Pi\) of all affine projectivities of a line onto itself or the group \(\Sigma\) of all projectivities of a line onto itself on the corresponding affine or projective plane has been studied for long. \textit{A. Schleiermacher} [Math. Z. 114, 313-320 (1970; Zbl 0182.231)] proved that \(\Pi\) or the stabilizer \(\Sigma_ p\) of a point p contains a regular normal subgroup if and only if the plane under consideration is a translation plane or a Moufang plane respectively. In a previous paper [Math. Z. 155, 11-21 (1977; Zbl 0354.50004)] the author deduced from this regularity condition that a locally compact connected affine plane is a translation plane if \(\Pi\) is locally compact in the compact-open topology (and hence \(\Pi\) is comparatively small). Weakening Schleiermacher's regularity condition or imposing topological conditions on the closure of \(\Pi\) in the group of all homeomorphisms of a line onto itself the author obtains various characterizations of desarguesian planes in the class of ordered planes and in the class of locally compact connected affine planes. In analogy to the description of the group \(\Sigma\) for projective Moulton planes of \textit{D. Betten} [J. Geom. 13, 197-209 (1979; Zbl 0426.51009)] the author shows in a short supplement that the group \(\Pi\) of an affine Moulton plane is isomorphic to the transformation group of all piecewise affine linear bijections of \({\mathbb{R}}\).
scientific article; zbMATH DE number 4009209
scientific article