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 geometry with automorphism group G = 22n · Ln(2) and covered by the truncated Coxeter complex of type D2n . The non-canonical ways give us geometries with smaller automorphism group (G ≤ 22n · (2n − 1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.