By L. Bers, I. Kra
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Additional info for Crash Course on Kleinian Groups
HI(F1,-~2q_2 ) The s t r u c t u r e of H I ( F , - ~ 2 q _ 2 ) d e p e n d s on t h e g e o m e t r i c m a n n e r i n w h i c h F is a s u b g r o u p of the full M~Sbius g r o u p . H o w e v e r , one c a n find a b o u n d on d i m H I ( F , ] - [ 2 q _ 2 ) t e r m s of the n u m b e r of g e n e r a t o r s L e m m a 3. Suppose Let vE T[2q_2 would mean that of F. q >_ 2 a n d I" is a n o n e l e m e n t a r y K l e i n i a n g r o u p generated by N elements. Proof: in Then dim HI(F,]~2q_2)<_ (2q-1)(N-1).
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.
Hence, has a potential F F0 such that v 0 be this is a potential w h i c h A = ~flI. by l e m m a 5, Thus, w e have, by /3 o i(~0) = 0. By ~ = 0. V. ,  L, Bets, Finitely generated 8__~6(1964), 413-429 An approximation Kleinian and 8__77(1965), theorem, groups, Amer. J. 759. J. , 14 (1965), I-4.  L. Bets, (1967),  L. Bets, Analyse OnAhlfors' finiteness theorem, Amer. J. , 89 113-134. , for finitely generated 18 (1967), 23-41. Kleinian groups, J. 47  L. Bets, Automorphic generated  I.
Crash Course on Kleinian Groups by L. Bers, I. Kra