By Lipman Bers (auth.), Lipman Bers, Irwin Kra (eds.)

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**Example text**

Type, I) = 0. with signature is a Riemann ) is injective. 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.

THEOREMS ( B e t s ' i s t a r e a t h e o r e m [4]) Area (Q/F)<_ 47r(N-l) where the a r e a is Poincar4 a r e a and N = the number of g e n e r a t o r s of F. Proof: - Merely multiply the inequality in corollary 2 to t h e o r e m 9 by 27r and let q-~ ~. q - The left hand side approaches the Poincar~ a r e a and the right hand side approaches 4Tr(N-1). 45 Theorem 11. ( B e r s ' 2nd a r e a t h e o r e m n o n e m p t y o p e n s u b s e t s of fi, suppose ~1 U f12 = fl and [4]) Suppose fll and ~2 e a c h of w h i c h i s i n v a r i a n t u n d e r fll N ~2 : ~" I f fil is c o n n e c t e d , are F, and then area(fl2/F) < area(fll/l~ ).

Is an open subset of a complex Banach space, Since M(G,E) it has a natural topology and complex structure. To every ~ E M(G,Z) a unique normalized there corresponds (Ahlfors-Bers (fixing 0,1,~) ~-conformal [4]) automorphism of denoted by w ~. 1. A Beltrami coefficient is called trivial if it satisfies followin$ conditions: a) ~ogo(w~)-i : g, all g ~ G, one (hence both) W E M(G,Z) of the 5O b) w~(z) = z, all z E A, the limit set of G. 2. of the following fixes each component elementary A__nnorientation preserving maps every component o_~f the complement of topological automorphism o f ~ of its fixed point set onto itself.