A Crash Course on Kleinian Groups: Lectures given at a by Lipman Bers (auth.), Lipman Bers, Irwin Kra (eds.)

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

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0, J F(z) F We satisfies ~(z) = 0 ( I z I 2 q - 4 ) , Moreover, therefore, It i s s i m p l e c o n s e q u e n c e 1/X (z) = 0(lz t2), H e n c e , f r o m the f a c t t h a t 0([C [-4) z -* ~. (see Kra ha I < ( e o n s t . ) x 2-q, we It f o l l o w s t h a t the i n t e g r a n d and the i n t e g r a l c o n v e r g e s a b s o l u t e l y f o r a l l the i n t e g r a l is 0([z-ajl becomes must Usingthis a b o v e , we m a y a s s u m e w i t h o u t l o s s of of S c h w a r z ' s I e m m a t h a t if = E f~, t h e n [6], p a g e 168).

Remark. It i s u n k n o w n w h e t h e r t h e a n a l o g o u s t h e o r e m valid for q > 2. 8. AHLFORS' ( A h l f o r s [1]) Kleinian group, 7 is S e e K r a [6]. §4. Theorem to theorem then FINITENESS If r fl(r)/r THEOREM is a finitely generated nonelementary is a finite union of Riemann surfaces of f i n i t e type. Remark. This theorem components. does not say that In m o s t c a s e s t h e n u m b e r ~(F) h a s f i n i t e l y m a n y of components of fi(F) i s infinite. T h e p r o o f of t h e f i n i t e n e s s classical Lemma theorem depends on the following lemma.

2q-1. log[z-ajl continuous show (2. 2). 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.

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