Some local 1 y

This paper is a continuation of Kantor (4, 6] and Ronan [10]. We will construct finite and infinite 'geometries' admitting flag-transitive groups and having buildings as universal covers. The first building is that of {1+ (6, (Ji2); the remaining ones are new and strange. As in the aforementioned papers, our motivation is the search for finite geometries that strongly resemble buildings, and for group-theoretic situations similar to BN-pairs. The paper is split into two independent parts. In section 2 we construct finite GABs (geometries that are almost buildings [4, 6]) with diagram n and rank 2 residues PG(2, 2). One of these is described for each odd integer m > 1. Each of them is covered by the {1+ (6, (Ji2) building. The GABs in Section 2 admit flag-transitive groups, and hence can be constructed as coset spaces exactly as in [6, sect. 2]. Consequently, all the relevant terminology is found in that reference (but compare 3.1 below). In Section 3 we will construct chamber systems (Tits [13], Ronan [9]) whose diagrams are complete graphs and whose rank 2 residues are PG(2, 2) or PG(2, 8). This section is relatively trivial. It is of interest only as an indication of the existence of pathological locally finite flag-transitive buildings. I am grateful to T. Meixner for his comments and corrections.