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

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Extra 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

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We fix once and for all a finitely generated elementary) (non- Kleinian group G, and let Z be an invariant of components for G. be denoted by ~. The region of discontinuity of G will All Kleinian ~rou~s under consideration be assumed finitely ~enerated. §i. QUASICONFORMAL DEFORMATIONS Let f be a quasiconformal will Almost all Fuchsian ~roups con- sidered will be finitely ~enerated~ ~Work partially union of the first kind. OF KLEINIAN GROUPS automorphism of the complex supported by NSF grant GP-19572.

Q Ultimately, /3 o i i s i n j e c t i v e f o r s p a n a d e n s e s u b s p a c e o f A (~) q when F q >_ 2. is infinitely q = 2 It i s n o t generated and q>2. 2. shown that By HI(F,T~2q_2 that it is the direct of Eichler Kra's introducing integrals. papers the space ) is the middle sum of the space For § 3. term B an exposition on this subject, see of Eichler q (Z,F) integrals, of an exact of this theory and has sequence and an appropriate and space a listing of [6]. AN APPROXIMATION THEOREM In t h i s s e c t i o n w e w i l l c o n c e n t r a t e o n t h e c a s e before, Kra q = 2.

To construct the mapping, ~(z) = k 2 - 2 q ( z ) ~ ( z ) for z E ~2 and be a potential for the generalized L(~)(z) 2w of c o h o m o l o g y g r o u p s by - q d2q-1 - F(z) d z 2 q -1 holomorphic for which potential By differentiating F that is c h o s e n , ~(z)= 0 for Beltrami z E ~1" f u n c t i o n in f12' l e t ¢p E Bq(fl 2, F) a n d l e t coefficient It i s c l e a r t h a t L(~) and that the integral expression z E C- Q2" Let F ~ and define L(~) will be a d e p e n d s o n l y on ~ a n d not on L is an a n t i l i n e a r m a p p i n g .

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