∗ k - geometries Antonio Pasini 1

Gluing two affine spaces

A construction is described in [2] by which, given two or more geometries of the same rank n, each equipped with a suitable parallelism giving rise to the same geometry at infinity, we can glue them together along their geometries at infinity, thus obtaininig a new geometry of rank n+ k− 1, k being the number of geometries we glue. In this paper we will examine a special case of that constructi...

Shrinkings, structures at infinity and affine expansions, with an application to c-extended P- and T-geometries

This paper is a shortened exposition of the theory of shrinkings, with particular emphasis on the relations between shrinkings and geometries at infinity or affine expansions. To make things easier, we shall only consider locally affine geometries, referring the reader to [A. Pasini, C. Wiedorn, Local parallelisms, shrinkings and geometries at infinity, in: A. Pasini (Ed.), Topics in Diagram Ge...

A tower of geometries related to the ternary Golay codes

On Flat Flag-Transitive c.c∗-Geometries

We study flat flag-transitive c.c∗-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometr...

A Generalization of a Construction due to Van Nypelseer

We give a construction leading to new geometries from Steiner systems or arbitrary rank two geometries. Starting with an arbitrary rank two residually connected geometry Γ, we obtain firm, residually connected, (IP )2 and flagtransitive geometries only if Γ is a thick linear space, the dual of a thick linear space or a (4, 3, 4)-gon. This construction is also used to produce a new firm and resi...