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By L. Bers, I. Kra

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X T(rr). 2 (Bers [ii]). holomorphic The group Mod(G,E) is a group of automorphisms o f ~ ( G , E ) . It acts properly 60 discontinuously o_n_n~(G,E), and ~(G,E) is thus a normal complex space. The group Mod(G,Z) is induced by quasiconformal auto- morphisms f of ~ that conjugate G into itself and fix E. There is thus a normal subgroup MOdo(G,E) of finite index in Mod (G,E) that is induced by quasiconformal automorphisms of A C that fix each E~~ = [g(Aj);g E G]. Let f be such an autoA morphism of $.

Q In particular, i f ~ ~ A, to be the closed subspaces Lq(A,F), respectively. is holomorphic as z~ Remark. of holomorphic When ~ E A, co t r a n s l a t e s B (A,F) q functions in and A (A,F) q L~(A, F) q and the condition that a function into the condition that %0 ~ ( z ) = 0( Iz 1-2q) ~. In t h e c a s e t h a t A (A) i n s t e a d o f q A2(U) at we define B (A,F) q F is the trivial group we write and coincides with the space A (A,F). q A(U) and A =U, the space by Earle in Lecture Ifwelet introduced B (h) q 2.

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).

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