Note di Matematica 24, n. 2, 2005, 97–133. Locally affine geometries of order 2 where shrinkings are affine expansions Antonio Pasini Department of Mathematics and Computer Science, University of Siena Pian Dei Mantellini, 53100 Siena, Italy pasini@unisi.it Received: 6/10/2004; accepted: 16/2/2005. Abstract. Given a locally affine geometry Γ of order 2 and a flag-transitive subgroup G ≤ Aut(Γ), suppose that the shrinkings of Γ are isomorphic to the affine expansion of the upper residue of a line of Γ by a homogeneous representation in a 2-group. We shall prove that, under certain hypotheses on the stabilizers Gp and G l of a point p and a line l, we have G = RGp for a representation group R of Res(p). We also show how to apply this result in the classification of flag-transitive c-extended P - and T -geometries. Keywords: shrinkings, affine expansions, representation groups, sporadic groups MSC 2000 classification: primary 51E24, secondary 20D08, 20C34 1 Introduction This paper is a continuation of a previous paper [14], by C. Wiedorn and myself. In [14], developing an idea of Stroth and Wiedorn [17] (but exploited also in [4], [9] and [8]) we built up a general theory of local parallelisms, geometries at infinity and shrinkings for geometries with string diagrams (called poset- geometries in [14]). We applied that theory to a number of examples taken from the literature, with special emphasis on the investigation of flag-transitive c-extensions of P - and T -geometries (Fukshansky and Wiedorn [3] and Stroth and Wiedorn [17]; see also Stroth and Wiedorn [18] for examples not considered in [3] and [17]). In particular, in Proposition 7.8 of [14] we put in full evidence the role that a combined analysis of shrinkings and structures at infinity had in [17]. However, by that method, we can only get control over c-extended P -geometries of rank n ≥ 4 where, by repeating the shrinking procedure n − 3 times, we end up with the c.P -geometry for 3 · S 6 , which has the tilde geometry as its structure at infinity. In all but one of these geometries the structures at infinity are T - geometries, whence known objects (see Ivanov and Shpectorov [6]). So, we can compare feasible geometries at infinity with feasible shrinkings. The latter have rank n − 1 and, if we work inductively, have already been classified at a previous step. In this way, one can classify the rank n case, too. In the remaining cases allowed by the hypotheses of [17] things go differently.