Extending locally truncated geometries

Given a set of types I. a type 0 is an element of I, a subset J of I containing 0, and a diagram I(0) over I{0}. a geometry Gamma over the set of types J is said to be locally truncated of I(0)-type if the J{0}-residues of Gamma are truncations of geometries or chamber systems belonging to I(0). We give a sufficent condition for such a geometry to be the J-truncation of a chamber system over the set of types I with all I{0}-residues belonging to I(0). Then we apply our result to some special cases. We exploit it to classify flag-transitive c(n),c*-geometries of rank n + 2 greater than or equal to 4 and c(n),c*-geometries of order 2. We give a new proof of a theorem of Ronan on C(n),L-geometries and we construct chamber systems for a number of sporadic groups in which certain well known geometries are involved as truncations or residues. (C) 2001 Academic Press.