Diagrams
A diagram is a structure defined on a set of types I. This structure generally is close to a labelled graph and provides information on the isomorphism class of residues of rank two of geometries over I. This way diagrams lead naturally to classification
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Diagrams
A diagram is a structure defined on a set of types I . This structure generally is close to a labelled graph and provides information on the isomorphism class of residues of rank two of geometries over I . This way diagrams lead naturally to classification questions like all residually connected geometries pertaining to a given diagram. In Sect. 2.1, we start with one of the most elementary kinds of diagrams, the digon diagram. In Sect. 2.2, we explore some parameters of bipartite graphs that help distinguish relevant isomorphism classes of rank two geometries. Projective and affine planes can be described in terms of these parameters, but we also discuss some other remarkable examples, such as generalized m-gons; for m = 3, these are projective planes. The full abstract definition of a diagram appears in Sect. 2.3. The core interest is in the case where all rank 2 geometries are generalized m-gons, in which case the diagrams involved are called Coxeter diagrams, the topic of Sect. 2.4. The significance of the axioms for geometries introduced via these diagrams becomes visible when we return to elements of a single kind, or, more generally to flags of a single type. The structure inherited from the geometry becomes visible through so-called shadows, studied in Sect. 2.5. Here, the key notion is that of a line space, where the lines are particular kinds of shadows. In order to construct flagtransitive geometries from groups with a given diagram, we need a special approach to diagrams for groups. This is carried out in Sect. 2.6. Finally in this chapter, a series of examples of a flag-transitive geometry belonging to a non-linear Coxeter diagram is given, all of whose proper residues are projective geometries.
2.1 The Digon Diagram of a Geometry The digon diagram is a slightly simpler structure than a diagram and is useful in view of the main result, Theorem 2.1.6, which allows us to conclude that certain elements belonging to a given residue are incident. Let I be a set of types. Definition 2.1.1 Suppose that i, j ∈ I are distinct. A geometry over {i, j } is called a generalized digon if each element of type i is incident with each element of type j . F. Buekenhout, A.M. Cohen, Diagram Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 57, DOI 10.1007/978-3-642-34453-4_2, © Springer-Verlag Berlin Heidelberg 2013
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Diagrams
Let Γ be a geometry over I . The digon diagram I(Γ ) of Γ is the graph whose vertex set is I and whose edges are the pairs {i, j } from I for which there is a residue of type {i, j } that is not a generalized digon. In other words, a generalized digon has a complete bipartite incidence graph. The choice of the name digon will become clear in Sect. 2.2, where the notion of generalized polygon is introduced. Example 2.1.2 In the examples of Sect. 1.1, the cube, the icosahedron, a polyhedron, a tessellation of E2 , and the Euclidean space E3 all have digon diagram ◦
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The digon diagram of Example 1.1.5 (tessell
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