Тик I a, Homeomorphic conjugates of Fuchsian groups

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Replacing X by X" we can assume that Eçy{A,X)==Q, Let C, and A, be as above. If £(y4, X) = 0, there is nothing to prove. Otherwise there is C, such that £(y4J>0. We now apply a similar reasoning as above. Since £о(Л) = 0, there is by Lemma 3E an axis В of Q equivalent to A^ and which is simple with respect to Gq. Hence В has at most binary intersection in G, and hence we can adjoin В to obtain Xß. It follows by Lemma 5B1 that Е{А,Хв)<Е{А,Х), and we obtain (3) after a finite number of steps. П

Suppose that E{A, Z) = 0. Let (C^,..., C„) be the component sequence of A in X and let v4, = ЛпС,. Since £(Л^ = 0, Lemma 3E implies that there is an axis В equivalent to A^ and which is simple with respect to the orientation preserving subgroup of GcJöCj. Hence, В has at most binary intersection with respect to Gc^\dC^ and consequently with respect to G. Thus, by Lemma 5A1, we can always adjoin an axis m the equivalence class of A^, provided that there is an A^. We keep on adjoining until there are no Л/s. This means that A becomes non-essential in X'. We will now define a number which is decreased in each extension and so ensures that the process ends.

Let A and В be multi-axes of X whose component sequences in X are (C^,..., C„) and {Z)i,..., Z)jt), respectively. We say that A and В intersect essentially at C, if the component sequences of A and В have in common exactly one element C^ such that AnC, and BnC, intersect essentially; note that even if A and В intersect, then the component sequences need not have common elements. Set

E' { A , X ) = k

if к is the number of g e G such that A and g A intersect essentially at some C,. Note that E'{A, X) is a notion different from E{A, X).

The fact that E'{A, X) is finite requires a small proof If it would be infinite, then there would be j and к and an infinite number of g, e G such that g,(5Cj) = öCfc and that A n Q and each g,(^ n Cj) intersect. Let a and b be the endpoints oi AnCy By the convergence property, we can pass to a subsequence so that gi{z)-^y for all zeX\L{G). Thus |g,(a)-g,(b)| —> 0 which is impossible since the endpoints of Л n Q are ordinary points. Hence E'{A, X) < oo.

We do not need so much the number E'{A, X) as the number

N { A , X ) = E'{A,X)-Vm

where m is the number of elements in the component sequence of Л in X (see (1).) If ЛГ(Л, X) = 0, then A is non-essential. By the next lemma, we can always find an extension of X so that this is true.

Lemma 5B3. Let A be a multi-axis of a G-complex X. Suppose that E(A, X) = 0 and that Ap = AnCp is an essential axis of dCp when P{A, X) = (Ci,..., С J. Let A' be an axis of Cp equivalent to Ap and which has at most binary intersection {with respect to Gc,\dCi) and let X' be obtained by adjoining A'. Then

N { A , X' ) <N { A , X ) ,

3i Journal fur Mathematik Band 391