  A lemma of Tits establishes a connection between the simple connectivity of
an incidence geometry and the universal completion of an amalgam induced by a
sufficiently transitive group of automorphisms of that geometry. In the present
paper, we generalize this lemma to intransitive geometries, thus opening the
door for numerous applications. We treat ourselves some amalgams related to
intransitive actions of finite orthogonal groups, as a first class of examples.
