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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**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**

**Example text**

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. 7. Let A be a component of ~(F) and let F 1 be the sub- group of F which leaves & invariant.

Now raise potential function (~,F). F k2-2q~ question: is is uniquely determined a2q_l}, Suppose for we know there exists Beltrami /~ o i an injective to asking whether the cohomology ~ (~), spana as c l a s s of t h e by the automorphic giving an answer, Z If t h e f u n c t i o n s z varies dense subspace /3 o i(~) = 0. - such that F(~) we prove over the set of Aq(5"), then F ff Then by lemma I A = 0. z In fact, 4, there F is a potential is given by /2-2q~- which 2-2q Z since both the integral vanish at the F(z) = 0 form H I ( F , TT2q_2 ) i s i n j e c t i v e .

Nature of F. signature Further, is a subgroup of finite The group Mod(g,n) (a classical (operating result, to G. ,~). equivalent classes only on the sig- to be Mod V for a group of If F has type (g,n) then Mod F index in Mod(g,n). acts d i s c o n t i n u o u s l y on T(g,n) for the proof see, for instance, [i0]) and thus the Riemann space Bets (for F) R(r) = ~(r,~) is a normal complex space (as a result of a theorem of Cartan 57 [14]). For more about the group Mod(g,n) see Royden's lecture. REMARK.

### 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 Thomas

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