By L. Bers, I. Kra

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0(llogIz-ajl that By F at ) as ) as z-~ a.. j and, Thus by letting a.. is continuous elementary everywhere estimates and one shows 32 F(z) = 0(]z]2q-2) as variable formula, one c a n s h o w that, constant C(R) z-b ~ and, by r o u t i n e u s e of t h e c h a n g e of for every R > 0 t h e r e is a such that I F ( z ) - F ( w ) [ < C(R) [ z - w I l o g l z - w l [ whenever lzl and twt < R. It remains tions. Let support. ) to show that 8F/Sz ~0 be a test function, We must show test function ~p. Let If F~o~ dz A d---z - = that is, a - ff that = ~ F~ in the sense C° function dz A d-'~ = ff ~ p(z) ff ~ - ~ ~(~) (~-z)p(~) with compact U~ dz A d---z for p(z) = (z-a l) • • ' (z-a2q_l).

Eoboundaries. HI(F,-~2q_2 ) ~ (B- 1¥B) = P ( y ) • B = B ~ _ q P ( y ) It i s e a s y to s e e t h a t t h e m a p p i n g preserves between Then for all A P - - > P is i n v e r t i b l e a n d It i s i m p o r t a n t to r e a l i z e t h a t if F 1 a n d 1"2 a r e K l e i n i a n g r o u p s a n d g : F 1 - - > 1"2 i s a n a l g e b r a i c i s o m o r p h i s m , t h e r e w i l l not, i n g e n e r a l , b e a n y r e l a t i o n s h i p b e t w e e n a n d H l ( F 2 , - ~ 2 q _ 2 ). HI(F1,-~2q_2 ) The s t r u c t u r e of H I ( F , - ~ 2 q _ 2 ) d e p e n d s on t h e g e o m e t r i c m a n n e r i n w h i c h F is a s u b g r o u p of the full M~Sbius g r o u p .

It acts properly 60 discontinuously o_n_n~(G,E), and ~(G,E) is thus a normal complex space. The group Mod(G,Z) is induced by quasiconformal auto- morphisms f of ~ that conjugate G into itself and fix E. There is thus a normal subgroup MOdo(G,E) of finite index in Mod (G,E) that is induced by quasiconformal automorphisms of A C that fix each E~~ = [g(Aj);g E G]. Let f be such an autoA morphism of $. For each j, there is a g~ ~ G such that fj = gjlof fixes Aj and fjGjfj I = Gj. By the introductory remarks of §6, there is an automorphism w.