Dihedral Closed Subsets

A closed subset of S is called dihedral if it is generated by a set of two involutions of S. 

In this chapter, we investigate dihedral closed subsets of S. The letter L stands for a set of two (different) involutions of S. We shall look at (L). As earlier, we shall write instead of tj, . 

In accordance with Section 3.5 we shall denote by S-i(L) the intersection of the two sets 5 i(Z) with l e L. 

It may happen that S-i(L) is empty. In this case, the structure of (L) is easy to describe. More challenging is it to look at the case where 5 i(L) is not empty, and it is this case on which we shall focus in the present chapter. In fact, we are mainly interested in the case where 5 i(L) contains exactly one element. 

Later in this chapter, in Section 10.6, we shall assume additionally that (L) has finite valency. It is one of the main goals of this chapter to show that (in this case) L is a Coxeter set (defined in Section 3.6) or a Moore set. (The definition of a Moore set will be given in Section 10.6.) This goal will be achieved in Theorem 10.6.6. 

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The results in this section show that the notion of a Coxeter set emerges naturally from the theory of dihedral closed subsets. On this occasion, it might be worthwhile to recall that the theory of Coxeter sets is equivalent to the theory of buildings in the sense of Jacques Tits. 

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