Geometry -- Von Staudt's Point of View: Proceedings of the NATO Advanced Study Institute Held at Bad Windsheim, West Germany, July 21--August 1,1980

Proceedings of the NATO Advanced Study Institute, Bad Windesheim, West Germany, July 21-August 1, 1980 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." Front Matter....Pages i-xi Projectivities In Projective Planes....Pages 1-49 Perspectivities in Circle Geometries....Pages 51-99 Cross-Ratios in Projective and Affine Planes....Pages 101-125 Cross Ratios and a Unifying Treatment of Von Staudt’s Notion of Reeller Zug....Pages 127-150 Projectivities in Free-Like Geometries....Pages 151-164 Existentially Closed Projective Planes....Pages 165-174 Conicoids: Conic-Like Figures in Non-Pappian Planes....Pages 175-196 Symmetries of Quadrics....Pages 197-229 Some New Results on Groups of Projectivities....Pages 231-248 Theorems About Reidemeister Conditions....Pages 249-273 Permutation Groups with Few Fixed Points....Pages 275-311 Projectivities and the Topology of Lines....Pages 313-337 Projectivities and the Geometric Structure of Topological Planes....Pages 339-372 Semimodular Locally Projective Lattices of Rank 4 from v.Staudt’s Point of View....Pages 373-400 The Impact of Von Staudt’s Foundations of Geometry....Pages 401-425 Back Matter....Pages 427-430