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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**Extra info 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**

If we set ~(G,z) = Mo(a,z),,,'fo(a,z), then it is clear that If E = ~, the entire ~(G) = ~ ( G , ~ ) , region of d i s c o n t i n u i t y ~(G) = ~ ( G , ~ ) , and Z(G) = $ ( G , ~ ) . §2. T}{E D E C O M P O S I T I O N Let ~ l , ~ 2 , . . , ~ r components in G. lecture) of finite THEOREM be a complete list of non-conjugate of E and Gj = G~ , the stability j By Ahlfors' Finiteness Theorem [3] subgroup Furthermore, of 4. J (see Gardiner's this is indeed a finite list and ~j/Gj type. we write is a surface each Gj is finitely generated.

Following theorem. Theorem g. B2(E, HI(F,-~2) F) has finite and these equals > spaces E/F), have is of finite Let of q/F F 8, we must To accomplish be a nonelementary is of finite type. thai Q(F)/F this, we prove Kleinian Then show group ~ o i :B q the such that each > (Q,F) ) is an injection. R Proof: Let w be a fundamental region in Q for F. w=U wi i=l where R F1 obviously /3 o i : 3, (which when 7. has finitely many HI(F,-~2q_2 A/F 1 example, of finite type. B2(A ,FI) = A2(5 ,FI) To finish the proof component surface A Therefore For Kleinian 5 be a single leaves we conclude finite dimension.

4). Furthermore, T(r)/Modi(r) ~ ~(G~,~), the subset of Both of these are groups. subgroup The group MOdH(F) in Mod r induced of ModH(F). acts freely on T(r). omorphic §~ universal coverin~ space T(r). THE GENERAL CASE We now consider the situation described in §2. j with covering group Hj c Fj. x MOdHrF r. 1 (Maskit [25]). x T(r ), and is hence simply connected. (a,Z). COROLLARY i. The deformation space ~(G,Z) is a complex analytic manifold with universal covering space ~(G,Z). COROLLARY 2. x T(rr).

### 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.)

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