By Lipman Bers (auth.), Lipman Bers, Irwin Kra (eds.)
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Additional resources for A Crash Course on Kleinian Groups: Lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco
41 This completes the proof. 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 ) Kleinian group, 7 is S e e K r a . §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.
An automorphism quasiconformal (This result has . For a rather We thus define T(g,n) (g,n). GROUPS e of the Kleinian with respect as a complex T(E) for some group r of type §5. MODULAR geometric, that T(r) ~ B(F) manifolds). Bers-Greenberg manifold, ~ B(r). group G is called to Z, if and only if there is a automorphism f of ~ compatible with G such that fZ = Z and e(V) = f~yof -I, all y ~ G. The map f induces a biholomorphic by sending ~ E M(G~Z) w~of-IIz. equivalence mapping into the Beltrami It is easy to check classes automorphism and hence that th~ induces of M(G,Z) coefficient mapping of preserves a biholomorphic self- e* : %(G,z) - %(G,z) w h i c h depends o n l y on the automorphism B, i n f a c t the conjugacy class We thus d e f i n e o f 8 modulo i n n e r automorphisms o f G.
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 ) 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 Crash Course on Kleinian Groups: Lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco by Lipman Bers (auth.), Lipman Bers, Irwin Kra (eds.)