Free constructions

Free extensions are often used in geometry to show the existence of models for a given theory and to construct examples whose properties are contrary to those of common models. Ever since Hall (1943) introduced free extensions for projective planes, the increasing interest for other classes of incidence geometries has led to repeated variations of Hall's ideas, where the constructions themselves only had to be adapted according to the different situations given by the language, and the axioms of the geometry under consideration. Besides a survey on significant results obtained thus far, the main purpose of this article is the development of a unifying treatment including all classes of incidence geometries which can be characterized by a set Σ of axioms formulated in a first-order language L (e.g., projective planes, affine planes, generalized n-gons, Benz planes, etc.). By using rather simple model-theoretic tools, we can define the notions of (hyper-)free, open, confined, closed, (hyper-)free extensions, and degenerate geometries without knowing Σ and L explicitly. To a vast extent, we succeed in reformulating and proving the main results concerning free extensions within this general frame. The application of these results reduces to an easy verification of some model-theoretic conditions on the axioms. Thus it becomes clear that the real nature of free extensions in fact lies beyond geometry. We hope that this insight will contribute to stop splitting research on that subject. On the other hand, our treatment yields new results, in particular, on the generalized n-gons. Surprisingly, groups of projectivities also prove themselves to be sensitive to our unifying point of view, as far as their algebraic structure as a free group or connection with the group of automorphisms is concerned (cf. Theorems 10, 11, and Theorem 15). In addition, repercussions arise about traditional definitions for the group of projectivities. So, for affine planes, central perspectivities with parallel carriers reveal themselves as natural as the usual parallel perspectivities (cf. Theorem 13). In the last section, in order to illustrate applicability of hyperfree extensions, we combine them with certain amalgamation techniques and broach three rather archetypical questions: Is every group G the full automorphism group of some model in a given class of geometries? May every model be embedded into some homogeneous model enjoying nice transitivity properties? For which classes of geometries do there exist highly transitive groups of projectivities?