
In Lect. Notes Math. 1181, 242-248 (1986; Zbl 0588.51015) the first author gave a construction for buildings of type M whose diagram contains no subdiagram of type \(A_ 3\), \(C_ 3\) or \(H_ 3\). The purpose of this paper is to extend the result to diagrams which may contain \(A_ 3\) or \(C_ 3\) subdiagrams but no \(H_ 3\) subdiagram. For doing this the authors introduce the concept of a blueprint. This is a set of data related to the several generalized n-gons which build up the building and contains information imposed by the diagram. For example in a diagram of type \(A_ 3\) the projective planes have to be Desarguesian. A blueprint is said to be realizable if there is a building carrying this blueprint. The first result of this paper says that a blueprint is realisable if its restriction to each rank 3 subdiagram is realisable. This result is used to show that there are buildings of any type. In fact the construction gives all buildings of spherical type, all buildings obtained from a BN- pair attached to Kac-Moody data (without using Lie-algebra) and many of the affine buildings.
Mathématiques, General theory of linear incidence geometry and projective geometries, 510.mathematics, Geometric group theory, chamber systems, buildings, Article, Other finite incidence structures (geometric aspects)
Mathématiques, General theory of linear incidence geometry and projective geometries, 510.mathematics, Geometric group theory, chamber systems, buildings, Article, Other finite incidence structures (geometric aspects)
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