Shadow geometries and simple connectedness

We generalize results by Tits [22], Rees [13] and Rinauro [14], proving for large classes of 2-simply connected geometries with string diagrams that being 'thin at top' is equivalent to being obtainable as shadow geometries from 2-simply connected geometries with 'broom' diagrams (defined in Section 1.3 of this paper) or with completely disconnected diagrams or with 'complete' diagrams (defined in Section 1.3 of this paper). A non-simple-connectedness criterion is also obtained, as a by-product of the above. We apply that criterion to some geometries for the groups S4(3), S6 (2), O′N, G2(2), Aut(HJ), Aug(G2(4)) and Aut(Suz), proving that they cannot be simply connected. The inquiry begun in this paper is continued in [12], where automorphism groups of shadow geometries are investigated.