\({\mathbf L}\cdot {\mathbf L}^*\)-geometries and \({\mathbf D}_n\)-buildings (Q1292851)

The main result of this paper is that every locally finite \(\text{PG} \cdot\text{PG}^*\)-geometry satisfying two geometrical conditions (LL) and (T), is a truncation of a \(\text{D}_n\)-building (by keeping only the elements of the natural \(\text{D}_3\)-subbuilding). A \(\text{PG} \cdot \text{PG}^*\)-geometry is a rank 3 geometry with linear diagram \(\text{PG} \cdot \text{PG}^*\), where a geometry with diagram PG consists of the points and lines (and natural incidence) of some projective space. It is called locally finite if the PG-residues are finite. Let the elements of such a geometry be called respectively points, lines and blocks. The condition (LL) says that two points are incident with at most one line, and condition (T) states that every triangle of points is incident with a block. This characterization of truncations of \(\text{D}_n\)-buildings is a very natural one. In fact, \textit{M. A. Ronan} [Proc. Lond. Math. Soc., III. Ser. 53, 385-406 (1986; Zbl 0643.51010)] characterizes (quotients of) \(\text{D}_j\)-truncations of \(\text{D}_n\)-geometries, \(4\leq j\leq n\). So the result of the paper under review can be seen as the finite case of the missing case \(j=3\) in Ronan's result (in the finite case no quotients arise). The author's proof consists of constructing a rank 4 geometry with truncation the original \(\text{PC} \cdot \text{PG}^*\)-geometry, and then apply Ronan's result. This is a rather straightforward thing to do, but the author generalizes her construction to locally finite \(\text{L} \cdot \text{PG}^*\)-geometries and there are many technicalities to be taken care of. The paper concludes with an interesting section on further problems related to the main results of the paper.