Journal of Lie Theory 20 (2010), No. 2, 311--327
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 2-closed, 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 2-closed 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 "q-closed geometry" on non-compact and orientable 2-manifolds of finite genus, we determine the manifolds admitting such geometries and we describe the q-closed geometries on them; among which are the classical ones on the plane.
Keywords: Transformation groups, foundations of geometry, q-closed geometry.
MSC: 37B05, 54H15; 51H05
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