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Authors: Hähl, Hermann
Title: Homologies and elations in compact, connected projective planes
Issue Date: 1981 Zeitschriftenartikel Topology and its application 12 (1981), S. 49-63. URL
Abstract: In a compact, connected topological projective plane, let Ω be a closed Lie subgroup of the group of all axial collineations with a fixed axis A. We compare the set З\A consisting of the centres of all non-identical homologies in Ω to orbits of the group Ω[A] of all elations contained in Ω and of its connected component θ = (Ω[A])1. It is shown that З\A is the union of at most countably many θ-orbits; moreover, З\A turns out to be a single θ-orbit whenever the connected component of Ω contains non-identical homologies. This result is analogous to a well-known theorem of André for finite planes. It has numerous consequences for the structure of collineation groups of compact, connected projective planes.
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