Shrinkings, structures at infinity and affine expansions, with an application to cc-extended PP- and TT-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 Geometry, in: Quaderni di Matematica, vol. 12, Second University of Naples, 2003, pp. 127–195] for a more general perspective. The following are our main sources: G. Stroth and C. Wiedorn [c-extensions of PP- and TT-geometries, J. Combin. Theory Ser. A 93 (2001) 261–280] and A. Pasini and C. Wiedorn [Local parallelisms, shrinkings and geometries at infinity, in: A. Pasini (Ed.), Topics in Diagram Geometry, in: Quaderni di Matematica, vol. 12, Second University of Naples, 2003, pp. 127–195] for shrinkings and geometries at infinity, A. Pasini [Embeddings and expansions, Bull. Belg. Math. Soc.-Simon Stevin 10 (2003) 585–626] and G. Stroth and C. Wiedorn [c-extensions of PP- and TT-geometries — A survey of known examples, in: A. Pasini (Ed.), Topics in Diagram Geometry, in: Quaderni di Matematica, vol. 12, Second University of Naples, 2003, pp. 197–226] for affine expansions and A. Pasini [Locally affine geometries of order 2 where shrinkings are affine expansions, Note Mat. (in press)] for shrinkings in connection with affine expansions. In the final section of this paper we shall discuss applications of shrinkings to the classification of flag-transitive cc-extended PP- and TT-geometries.