By Wehrfritz B.A.F.

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For s e c o n d map b e i n ~ The s p a c e T ( F ) the Fuchsian a_~n i s o m o r p h i s m ) . is called the Teichm~ller space group F. B e f o r e we c a n c o n t i n u e m a p p i n g s , we m u s t t u r n the investigation t o a more c a r e f u l of the above examination of §4. THE FUCHSIAN CASE For r Fuchsian, we denote by B(r) = B2(F,L) space of bounded automorphie forms of weight support in the lower half plane, For ~ E M(F), let ~ the Banach -4 for F with L (see Gardiner's be the Schwarzian derivative lecture).

Is the loxodromic (ii). If (ii) h o l d s , = F(~) But such fixed points are dense in transformation (iii) holds, then each polynomial A, U has for every loxodromic with fixed point O n e concludes, by the continuity of F, vanishes on then ~, an infinite set, and therefore P (z) = 0. Y A then that F I h = 0. ). ) q are )(y(z))y'(z) q = ~(z) if we k2-q(z). holomorphic let ~ =k 2- Furthermore, and 2%, one such it is can easily # show that Theorem Yl-q, 5. 1 ~ = U, Consider and so u E M where i(~) = k2-q~ - and Proof: What generalized of i(~) must A q and z /~ > H I ( F , - ~ 2 q _ 2 ) is defined as before.

Since B(F) is a finite dimensional space, it can be shown using the implicit function theorem that ~(M(F)) = ~(T(F)) is open in B(F). 1 (Bers [8]). The Teichm~ller space T(F) has a unique complex structure so that : M(F) ~ T(F) i___ssholomorphic with local holomorphic sections. We can realize T(F) as a bounded domain (of holomorphy) i__%B(r). Further T(r) i_~stopolo~ically a cell. The fact that T(r) is a domain of holomorphy was proven by Bers-Ehrenpreiss [12]. ) Teichm~ller's theorem (see for example Ahlfors [2] or Bers [6]) implies a unique that every ~ £ M(F) Teichm~ller coefficient coefficient; is equivalent to that is, a Beltrami ~ with ~(z) = k ~(~)/I~(~)I , z ~ U, k ~ ~, 0 ~ k < l, It follows 0 /~ easily from this result (as topological depends been obtained simple proof [13] show that T(F), only on the type of r.

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