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

ISBN-10: 3540068406

ISBN-13: 9783540068402

ISBN-10: 354037776X

ISBN-13: 9783540377764

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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. 7. Let A be a component of ~(F) and let F 1 be the sub- group of F which leaves & invariant. Then A2(&,F I) has finite dimension if and only if &/F 1 is of finite type. Proof: By classical can construct function theory an infinite dimensional abelian differentials on ( s e e K r a [6], p a g e 3 2 4 - 3 2 8 ) o n e s p a c e of s q u a r e integrable S = A/F 1 if S has infinite genus or if S has infinitely many elliptic points or punctures.

Remark. ]Beltrami A for for z-* ¢o. * = Al_q, A * F = AI_qF If A is a M~Sbius coefficient i~ for F, is a generalized is a potential for transformation, and F a potential ]Beltrami A Li. ~ a generalized for coefficient ~, for then A IFA and 31 Lemma 4. If ~ nonelementary is a generalized Kleinian group a I, a2, .... 3) F(z) = is a potential Beltrami F with s u p p o r t in Z, are distinct points in A(F), for the a n d if then ~t(~)d~ A d~ (~-z)({-a I) ... (C-a2q_l) ( z - a 1) ,.. ( z - a 2 q _ l ) f f 2wi Z for coefficient U.

Of Kleinian > GA > [i} and F Fuchsian~ (as a m a t t e r of fact~ groups and F and GA U/~ ~ A/G A as surfaces with ramification points). 53 REMARK. 1) is [25]. 1). we define h*~ ~ M(G~,~) by and verify almost trivially that h~ : M(r) ~ M ( % , ~ ) a (linear, whenever possible) isomorphism. 1. 1) induces holomorphic surjections T(r) = ~ ( r , ~ ) ~ ~(%,~) T(F) ~ ~ ( G a , ~ ) (with the REMARK. for s e c o n d map b e i n ~ The s p a c e T ( F ) the Fuchsian a_~n i s o m o r p h i s m ) . is called the Teichm~ller space group F.

### 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.)

by Daniel

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