. Proof, The fact that X is an F -maximal bifix code of F -degree d results from Corollary 6.2.3. Let us show that G F (X) = G(Z) Let B = (R, 1, 1) be the minimal automaton of Z *

A. Let, Denote by Im(w) the image of ?(w) with respect to A Thus Im(w) = {t ? Q | s · w = t for some s ? Q}. Let u ? F be a word with d parses with respect to X. Let I = Im(u) By Lemma 7.1.4, the word u has rank d and thus Card(I) = d. Let Y = R F (u) be the set of first return words to u. By Theorem 6.5.2, the set Y is a basis of the free group A ? . For any y ? Y , the restriction of ?(y) to I is a permutation of I. Indeed, uy ? A + u implies Im(uy) ? I. Since uy ? F , the set Im(uy) has d elements by Lemma 7.1.5 The restriction of e to I is the identity, Thus Im(uy) = I. Since Im(u) = I, this proves the claim. Let e be an idempotent in ?(Y + )

G. Let, be the maximal group contained in ?(A * ) which contains e. It is a permutation group on I which is equivalent to G F (X)

?. For-y, let ?(y) be the restriction of ?(y) to the set I. For any y ? Y * , e?(y)e has the same nuclear equivalence and the same image as e. By Proposition 7.1.2 it implies that they are in the same H-class

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