By L. Bers, I. Kra

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27r---i(C-z)(C-al)(~-a2)(~-a3) = where a I, a 2, varies over the set We when q = 2, a3 have are three A - [a I, seen it suffices distinct fixed points a 2,a 3} A and z g that in order to prove in to prove the following that ~ o i is injective theorem. 37 Theorem 7. ( B e r s [2]) Kleinian group I', Let fl b__~eth__eelimi_____tse___ttof a n o n e l e m e n t a r V (I" m a y b e i n f i n i t e l y g e n e r a t e d ) . 2,a3] ~z(C) w h e r e z 6 h - [a 1,a Remarks O b v i o u s l y the t h e o r e m 1.

Dimension, sphere of Q/F. I) = B2(E,F). HI(F,-~2 A2(A,F be the union of all components component the subgroup dim in 3). is a finitely generated, of Q (F)/F Let (cf. Theorem for this dimension is a positive integer or R = and each 7r(wi) yields 43 precisely one of the components /3 o i(~) = 0. By lemma of fi/F. X2-2q~-- h a s a p o t e n t i a l ¥ E F. _Oi(z) = t~(z) if z E F~u. (z) = 0 o t h e r w i s e . We must 1 i. [,ilak2-2q[dz A dz I = f f * i 3 - ~ - Idz ^ ¢i goi =ff ~-g (F,i)[dz A Tzl gO.

B e r s [2]) Kleinian group I', Let fl b__~eth__eelimi_____tse___ttof a n o n e l e m e n t a r V (I" m a y b e i n f i n i t e l y g e n e r a t e d ) . 2,a3] ~z(C) w h e r e z 6 h - [a 1,a Remarks O b v i o u s l y the t h e o r e m 1. a r e d e n s e in A2(E) w h e r e b e c a u s e in t h i s c a s e 2. T h e n the f u n c t i o n s s p a n a d e n s e s u b s p a c e of A2(fi). i m p l i e s t h a t the f u n c t i o n s tz(~) E i s any u n i o n of c o m p o n e n t s of ~, A2(E) c A2(i~ ).