
Journal of Lie Theory 20 (2010), No. 2, 311327 Copyright Heldermann Verlag 2010 On "Axiom III" of Hilbert's Foundation of Geometries via Transformation Groups Athanassios Strantzalos Dept. of Mathematics, National and Kapodistrian University, Panepistimiopolis, Athens 15784, Greece astrantzalos@gmail.com Polychronis Strantzalos Dept. of Mathematics, National and Kapodistrian University, Panepistimiopolis, Athens 15784, Greece pstrantz@math.uoa.gr In 1902, D. Hilbert presented a foundation of classical plane geometries based on three topological axioms concerning a group G of homeomorphisms of the real plane. The third of these axioms required essentially that the action of G on the plane be 2closed, thus ensuring a kind of compatibility between the topological and the geometrical (in Klein's spirit) structures of the plane. In the present paper we show that the 2closed actions on noncompact, connected, locally connected and locally compact spaces are essentially restrictions in dense (eventually not strict) subgroups of groups acting properly on the considered spaces. Generalizing Hilbert's setting, we define the notion of a "qclosed geometry" on noncompact and orientable 2manifolds of finite genus, we determine the manifolds admitting such geometries and we describe the qclosed geometries on them; among which are the classical ones on the plane. Keywords: Transformation groups, foundations of geometry, qclosed geometry. MSC: 37B05, 54H15; 51H05 [ Fulltextpdf (209 KB)] for subscribers only. 