Sechzehndimensionale lokalkompakte Translationsebenen, deren Kollineationsgruppe G2 enth�lt

Abstract

New results of Salzmann and Hubig say that a 16-dimensional (locally) compact topological projective plane in which the group \(\mathbb{G}\) of continuous collineations has dimension ≥40 is a translation plane. It is therefore important to determine all 16-dimensional locally compact translation planes with dim \(\mathbb{G}\)≥40. From previous work of the author ([10]), it is known that such a plane is either the classical octonion plane, or dim \(\mathbb{G}\)=40 and \(\mathbb{G}\) contains a subgroup isomorphic to the compact exceptional group G2, but no larger compact simple subgroup. In the present paper, all planes satisfying the latter property more generally with dim \(\mathbb{G}\)≥38 are explicitly determined. Together with the classification of all 16-dimensional locally compact translation planes in which \(\mathbb{G}\) contains Spin(7) given by the author in [8], one thus knows all 16-dimensional locally compact translation planes with \(\mathbb{G}\) containing G2 and dim \(\mathbb{G}\)≥38. Via suitable Baer subplanes, the classification makes use of analogous results for 8-dimensional planes ([7]).