Shadow geometries and simple connectedness (Q1315379)

This interesting paper gives a description -- in a very general and axiomatized setting -- of the connection between large classes of ``weak geometries'' (i.e. geometries belonging to a Buekenhout diagram of some special form where the parameters corresponding to certain nodes can be equal to 1) and thick ones. In fact, the paper proves that under certain -- weak -- conditions on the geometry and -- stronger -- conditions on the diagram, one can ``turn a weak geometry into a thick one'' and vice versa, changing the diagram of course. The standard example here is any polar space of type \(D_ n\), which can be viewed as a weak polar space of type \(C_ n\). But the author's result applies to many other diagrams, including many Coxeter diagrams, and hence it can be applied to a large of buildings. It should be noted that along the way, the author establishes a sufficient condition to conclude that a given geometry is not simply 2- connected. This is applied to geometries arising from some simple (including some sporadic) groups and explicit examples are given.