On "Axiom III" of Hilbert's Foundation of Geometries via Transformation Groups

Athanassios Strantzalos¹ ; Polychronis Strantzalos¹

¹Dept. of Mathematics, National and Kapodistrian University, Panepistimiopolis, Athens 15784, Greece

Journal of Lie Theory, Tome 20 (2010) no. 2, pp. 311-327

Résumé

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.

Détail

Classification : 37B05, 54H15, 51H05
Mots-clés : Transformation groups, foundations of geometry, q-closed geometry