Extending Locally Truncated Geometries

Given a set of types I, a type 0∈I, a subset J of I containing 0, and a diagram I0 over I\{0}, a geometry Γ over the set of types J is said to be locally truncated of I0-type if the J\{0}-residues of Γ are truncations of geometries or chamber systems belonging to I0. 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 I0. Then we apply our result to some special cases. We exploit it to classify flag-transitive cn.c*-geometries of rank n+2⩾4 and cn.c*-geometries of order 2. We give a new proof of a theorem of Ronan on Cn.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.