Basic and degenerate pregeometries

We study pairs (Γ ,G), where Γ is a ‘Buekenhout–Tits’ pregeometry with all rank 2 truncations connected, and G 6 AutΓ is transitive on the set of elements of each type. The family of such pairs is closed under forming quotients with respect to G-invariant typerefining partitions of the element set of Γ . We identify the ‘basic’ pairs (those that admit no non-degenerate quotients), and show, by studying quotients and direct decompositions, that the study of basic pregeometries reduces to examining those where the group G is faithful and primitive on the set of elements of each type. We also study the special case of normal quotients, wherewe take quotients with respect to the orbits of a normal subgroup of G. There is a similar reduction for normal-basic pregeometries to those where G is faithful and quasiprimitive on the set of elements of each type. © 2012 Elsevier Ltd. All rights reserved.