Ever since F. Klein designed his "Erlanger programm", geometries have been studied in close connection with their groups of automorphisms. It must be admitted that the presence of a large automorphismgroup does not always have strong implications for the incidence-th- retical behaviour of a geometry. For exampl~ O. H. Kegel and A. Schleiermacher [Geometriae Dedicata 2, 379 - 395 (1974)J constructed a projective plane with a transitive action of its collineation group on quadrangles, in which, nevertheless every four points generate a free subplane. However, there are several important special classes of geometries, in which strong implications are present. For instance, every finite projective plane with a doubly transitive collineation group is pappian (Theorem of Ostrom-Wagner), and every compact connected projective plane with a flag-transitive group of continuous collineations is a Moufang plane (H. Salzmann, Pac. J. Math. ~, 217 - 234 (1975)]. Klein's point of view has been very useful for numerous incidence structures and has established an intimate connection between group theory and geometry vii P. Plaumann and K. Strambach (eds. ), Geometry - von Staudt's Point of View, vii-xi. Copyright © 1981 by D. Reidel Publishing Company. viii PREFACE 1. 1:1ich is a guidepost for every modern t:reat:ment of geometry. A few decades earlier than Klein's proposal, K. G. Ch. von Staudt stated a theorem which indicates a different point of view and is nowadays sometimes called the "Fundamental Theorem of Projective Geometry".
I: General Theory.- Projectivities in Projective Planes.- Perspectivities in Circle Geometries.- Cross-ratios in Projective and Affine Planes.- Cross Ratios and a Unifying Treatment of Von Staudt’s Notion of Reeller Zug.- Projectivities in Free-like Geometries.- Existentially Closed Projective Planes.- II: Projectivities and Conics.- Conicoids: Conic-like Figures in Non-Pappian Planes.- Symmetries of Quadrics.- III: Projectivities in Special Geometries.- Some New Results on Groups of Projectivities.- Theorems about Reidemeister Conditions.- Permutation Groups with Few Fixed Points.- Projectivities and the Topology of Lines.- Projectivities and the Geometric Structure of Topological Planes.- Semimodular Locally Projective Lattices of Rank 4 from v. Staudt’s Point of View.- The Impact of Von Staudt’s Foundations of Geometry.- Index of Subjects.
