Faithfully T-embedded

Theorem 6.2.8 LetT and U be closed subsets of S such thatT C U. Assume that U contains a closed subset V of S which covers T. Assume further that h G U 1 ThT C V and that V is faithfully T-embedded in S. Then, if U//T is faithfully embedded in S//T, U is faithfully embedded in S. 

In the last of the seven sections of this chapter, we investigate closed subsets T of S which have finite valency and satisfy O 1 (T) C 0 (T). We shall give a sufficient criterion for T to be faithfully embedded in S. 

Proposition 6.2.2 says that the property of being faithfully embedded is inherited to closed subsets. Given closed subsets T and U in S with T C U, we shall now ask ourselves under which hypotheses the property of being faithfully embedded in S can be lifted from T to U. 

Let us now assume that, for each element x in X, the Schur group G of T with respect to x acts transitively on xT and that, for each element t in T, Gx acts transitively on xt. We have to show that T is faithfully embedded in T. 

We shall prove that, under a certain additional condition, T is faithfully embedded in S. ... 

Recall that L is called a Coxeter set if L is constrained and satisfies the exchange condition. Recall also that L is called spherical if S- (L) is not empty. Recall, finally, that a closed subset T of S is called faithfully embedded in S if, for any two elements y in X and z in yT, each faithful map x from y, z to X extends to a bijective map from yT to yyT. 

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