Embedding geometric lattices in a projective space

Summary

INTRODUCTION

We have obtained a necessary and sufficient condition for a geometry to be embeddable in a (generalized) projective space. This condition essentially requires that all intervals above a point are embeddable in such a space and that these embeddings are “compatible” along the lines of the geometry.

All geometries considered here are geometric lattices; embeddings are isometric in Kantor's sense (see [14]).

This result provides a unique general proof of the following known theorems on locally embeddable geometries of dimension at least four: Mäurer's result on locally affine geometries (the socalled Möbius spaces), Kantor's strong embedding theorem and his other embedding theorems involving universal properties (see [16] and [13], [14]). The result can be extended to the dimension three case, generalizing Kahn's work on Mobius, Laguerre and Minkowski planes (see [8] to [11]).

Let us mention one of its new applications: Kantor's concept of strong embedding can be extended to a larger class of geometries; for instance, it is possible to prove the embeddability of infinite locally affino-projective geometries of dimension at least four.

The notions of geometry and embedding are defined in §2 and §3 respectively. The main result can be found in §4, where more historical details and an idea of the proof are given. This proof is based on a general embedding lemma stated in §5. Finally, §6 presents some applications; it also gives a precise connection between the main result in §4 and the known theorems mentioned above.