f f Dla~]~F~ ldxdy< oo

(8.13)

h~ = w,o(w~)-l: D-~ U (8.14)

(cf. the proof of L e m m a 3.1) we define a function ~Fo(z),

zE U-Av,

b y the requirement

Wo(h(z))h'(z) 2 =~F~

in view of (8.13) the definition is legitimate. This function is holomorphic in

U-A,,

has simple poles at all points of A~, and, since

h~G~hjl=G~,

we have

~t~o(g(z))g'(z) ~

=~Fo(z ) for

geG,,

(8.15)

f f {~o(z)]dxdy< c~.

(8.16)

U/G v

F r o m (8.15) and (8.16) one concludes in the usual way (using Fourier series for

~F o near parabolic fixed points of G~) t h a t in every fundamental domain for G~ in U the function

ly-~o(Z)l

is uniformly bounded except near a point of A,. This implies t h a t

f f U/G ~(Z)-~

I~FO (Z)I s

dx dy< oo.

(8.17) Hence we can find numbers cl, c 2 .. . . . c r and c 4= 0, such t h a t the function

l i f t + 1 (Z) = C1~2 0 (Z) + CC 11~21 (Z) + . . . + CCr~ r (Z) (8.18) satisfies the relations

~ / ~(z)-2~Fr+l(z)~F~(z)dxdy=O, ]=1,2 ... r,

(8.19) U/G~,

a n d f ( ,~(z)-2]~'Zr+l (z)[2dxdy= 1. (8.20)

J J U/G If (~1 . . . ~r+l) E C r+l, set

a(z) = 2(z) -~ [$1 ~F1 (z) + . . . § ~ r ~ r (Z) + ~+1 ~Fr+l (z)] (8.21) and note t h a t

aELoo(U, G~).

We define an element

TELoo(U,

G)I, for sufficiently small values of ]$1] ~+... + ]r b y the requirement t h a t

FIBER SPACES OYER TEICH~IULLER SPACES 119

w~ = woow~ (8.22)

a n d consider the m a p p i n g

(~1, .,, ~+i) ~-> ([w~], w~(a)) (8.23) of a n e i g h b o r h o o d of t h e origin in C ~+1 into Fu(G); t h e i m a g e of t h e origin u n d e r this m a p p i n g is ([wr], w~(a)); Since (r d e p e n d s h o l o m o r p h i c a l l y on (~1 ... ~r+i) so do v a n d w~:(a).

T h u s t h e m a p p i n g (8.23) is holomorphic.

L~MMA 8.4. The rank o/ the mapping (8.23) at ~i . . . ~ + i = O is r + l .

Proo/. L e t (t i ... tr) be t h e s e m i - s t a n d a r d coordinates in T(G) a b o u t t h e p o i n t [w~], belonging to t h e p o i n t wv(a), a n d let tr+i be a c o m p l e x n u m b e r restricted to t h e neighbor- h o o d of wv(a). T h e m a p p i n g (8.23) defines t i ... tr a n d tr+ 1 =w*(a) as h o l o m o r p h i c functions of ~i ... ~r+i defined n e a r t h e origin; we m u s t show t h a t

B y construction

~(tl . . . #+1)

a(r

. . . ~+~)

a~k if ] = k a n d it will suffice to show t h a t

# 0 a t r

for l<<,~,k<<,r at ~ l = . . . = $ ~ + i = O ,

~ r + i 0 for ] = 1 . . . r; ~ i = , . - = ~ + i = 0 (8.24)

~ t r + l

a n d ~ r + i # 0 f o r ~l = . . . = ~r+i = 0. (8.25)

Define a0 E Lor (U, G~) b y

a0 (z) = 2(z) -2 ~Fr+l (z). (8.26)

C o n d i t i o n (8.24) is e q u i v a l e n t to t h e r e q u i r e m e n t t h a t a0 be locally trivial. This is so, in view of L e m m a 8.1 a n d (8.19).

N o w define, for e E@, [e[ sufficiently small, t h e e l e m e n t T~ EL,~(U, G) b y t h e require- m e n t

w~ = W~oOW~.

A well-known (formal b u t legitimate) calculation of t h e B e l t r a m i coefficient T~ yields

e&o + v ~w, (z)/a5

where bo(z) = aoOW,(Z) - - (8.27)

T~- 1 + e&o v ~w, (z)/&"

Condition (8.25) is e q u i v a l e n t to t h e following:

~w*~(a) ~= 0 for e = O,

&

120 or, setting t o

L I P M A N B E R S

/8 = w TS~ (w~) -1,

8/~(w~(a))~O for e = O . (8.28)

8e

N o w / 8 is a quasiconformal a u t o m o r p h i s m of ~ which leaves O, 1, co fixed. L e t 0 be t h e B e l t r a m i coefficient o f / ~ ,

~1./~'

t h e n 01C - D = 0

a n d one computes, using (8.14) a n d (8.27), t h a t for zE D,

O(z) = ~0o (z), Oo (z) = aooh~ (z) g ( z ) / g (z).

B y a k n o w n f o r m u l a (cf. [18], p. 133 or [2], p. 104) 8]~(w~(a)) = l ( ( Oo(z) dxdy

_ 1 ~ ,~(h~ (z)) -~ ~,+~ (h. (z)) h'. (z)/h; (z) dx dy x~ JJD Z(Z-- l) [Z -- W'(Z)]

1 ~ f f g O0(Z) dx dy

where A is a s m o o t h l y b o u n d e d f u n d a m e n t a l d o m a i n for G ~ in D. Since 00 (g(z)) (g'(z)/g'(z) = 00 (z) for g E G ~, a simple calculation yields

8/~(w~(a)) ~

~s ~+o = O~ ~F~ (z) dx dy = ao (h~ (z)) q2" o (h~ (z)) ]h'~ (z)12 dx dy

j j U. Gv C

w h e r e we used (8.26), (8.18), (8.19) a n d (8.20). T h u s (8.28) is p r o v e d a n d so is L e m m a 8.4.

T h e l e m m a implies t h a t ~t ... ~r+1 can be used as c o m p l e x coordinates in n e i g h b o r h o o d of t h e p o i n t ([wv] , w"(a)) in F,(G). W e call t h e m standard coordinates a b o u t this point.

LwMMA 8.5. The standard coordinates about a point in Fu(G) are uniquely determined, except that the/irst r coordinates may be subject to a unitary trans/ormation and the (r + 1)-st may be multiplied by a complex number o/absolute value 1.

This follows f r o m t h e construction.

N o w we are in a position t o s t a t e

FIBER SPACES OVER TEICttMITLLER SPACES 121 LE~MA 8.6.

Under the hypotheses o/ Theorem 9, and /or

dim

T(G)=r< oo, the sur- jection T(G)~ F~ ( G) o/ Lemma 6.7 takes (appropriately chosen) standard coordinates about any point in T(O) into standard coordinates about the correspondinq point in Fu(G).

Proo/.

Let

#EL~(U,

0)1 and v=0(~u), Q being the mapping defined in w 6.5. Let tF 1 ... tFT+ 1 be the functions used to define the standard coordinates in $'u(G) about ([w~],

w~(a))

and let v~:

U ~ U - A ~ = w ~ ( U - A )

be the covering map in diagram (6.7).

Then we have t h a t

~(v.(~)) I v'~(~)l= 1-.

Next, define the functions Y

v~(z), ]=1 ... r+l.

One verifies by a computation t h a t the r belong to B ~ ( U , ~ ) where

G,=w,~w~ 1,

and t h a t

f f y2~j(z)~Fk(z)dxdy={ ~

if j=~k

v/~t, if ?" = k.

Let $1 ... ~r+l be complex numbers with small absolute values, let a and v be defined b y (8.21) and b y (8.22), respectively, and define

OELoo(U, ~)~

and

~ELo~(U, ~)1

b y the rela- tions:

~(~) = y~E~Cr~ (~) + ... + ~,+~cr,+l (~)],

and w~ = w~ ow,.

There is a universal covering ~:

U~w~owv(U-A)

and a commutative diagram

U w~ , U

w.(U- A ) , w.ow~(U- A) l

(8.29)

which shows t h a t

This proves the assertion.

V _ _

vl

U - A

w~ ~ U

,w~(U-A)

=0(~).

Lemma 8.6 contains the desired local description.

this is established in the same way as diagram (6.7) in the proof of L e m m a 6.5. Diagram (8.29) and (6.7) yield the commutative diagram

122 LIPMAN BERS

w 9. Direct proof of the isomorphism theorem

We shall now establish Theorem 9 without assuming previous knowledge of the topological result stated as L e m m a 6.9. The proof will also yield this lemma, for the case when the fundamental group of the surface S is finitely generated.

We begin b y proving, as before, L e m m a s 6.1-6.8. N e x t we consider the case d i m T(G) < ~ . L e t v EL~o (U, G)I, and note t h a t the preimages of the point ([wv] w ~ (a)) E Fu(G) under the m a p p i n g T(G)-+Fu(G) form a discrete set. Indeed, if #, p eL~(U, d)l are such t h a t ~(/~)=~(fe)=v, cf. L e m m a 6.5, then the R i e m a n n surfaces U/w~Gw~ 1 and U/w~Gw9 are both conformally equivalent to w , ( U - A ) / w , Gw~ 1, so t h a t w~Gw~ 1 and w~Gw~ 1 are conjugate in Qoo~. Thus [w~] and [w~] are equivalent under the group Mod (d), which acts discontinuously on T(G). The discreteness of every fiber of T(~)-~ F~(G), together with L e m m a 8.6 i m p l y t h a t T(G)~Fn(G ) is an unbounded unrami/ied covering.

Now, F~(G) is homeomorphic to T(G)• U, b y L e m m a 6.4. Since dim T(G)< ~ , T(G) is a cell, b y a classical theorem of Fricke. I n our ease, G without torsion, this also follows from the theory of extremal quasiconformal m a p p i n g (cf., for instances [6]).

We conclude t h a t Fu(G ) is a cell, so t h a t T(G)~Fn(G ) is an isomorphism.

Now we can prove L e m m a 6.9 assuming t h a t S is compact, except perhaps/or/initely many punctures. There is no loss of generality in assuming t h a t the a u t o m o r p h i s m 0 of S, which leaves the point d fixed, is quasiconformal. Indeed, in view of our hypothesis on S, O I S - ( d } is homotopic to a quasiconformal automorphism. We m a y also assume t h a t S = u/a, G a torsion free Fuchsian group, and d is the image of some a E U under U--->U/G. I n view of our assumption on S, dim T ( G ) < ~ . There is an eoEN(G) such t h a t the diagram

(D

U , U

i o i

u/G , u / e

(9.1)

commutes. L e t Or, 0~<t~<l be the h o m o t o p y which takes 0 into the identity leaving d fixed. I t lifts, for each t, to a m a p p i n g ~ t such t h a t we have a c o m m u t a t i v e diagram

U , U

u/G , u / a

(9.2)

with | ~ 0 = w , 0 1 = i d ,

FIBER SPACES OVER TEICKIV~I'JLLER SPACES 123

~ t o g o ~ - I E G (9.3)

and ~t(a) c A (9.4)

where A is, as in w 6, the G orbit of a. For t = 1, the commutativity of (9.2) implies t h a t

~IEG. B u t the commutativity of (9.1) defines o) only up to premultiplication b y an element of G. Hence we m a y replace co b y ~-1oco, and ~ t b y ~ f l o ~ t , i.e., we m a y assume t h a t

~ = id. (9.5)

Observe t h a t

~t(z)

depends continuously on (t, z)E [0, 1] • U. Since the group G is discrete,

Fttogo~ 1

depends only on

gEG,

and not on t. B y (9.5), we have t h a t

wogoo -1 =g

for all g E G. Since G is of the first kind, we conclude that [r = [id]. Hence (o EQnom and there is a

vELoo(U, G)I

with eo=[wp], [wp]=[id].

Since A is discrete,

~t(a)

in (9.4) does not depend on t; b y (9.5) we have t h a t

~t(a) =a;

for t = 0 , this yields

w~(a)=a.

Since [wv]=[id] implies t h a t

wv=w ~

we conclude t h a t

w~(a) =a.

Thus, for G as in Theorem 9, we have t h a t

T(O) ~ Fu(G)

takes [id] into ([wv],

w~(a)).

(9.6)

Now let /~

=@--1(~)) ELoo(U, 0)1,

(9.7)

cf. Lemma 6.5. Then we have the commutative diagram

U W/~ , U

vl

w, l U - A

U - A , U - A

I O[(U_A)/G 1

( U - A)/G , ( U - A)/G

(9.8)

obtained b y combining (6.7) and (9.1), noting t h a t (o =wp,

wv(A )

= A , and restricting the latter diagram.

Since b y (9.7) and L e m m a 6.7,

T(d)~F,,(G)

takes [w~] into ([w~],

w~(a))

and the mapping is known to be a bijection, (9.6) implies t h a t [w~] = [id]. Hence there is an Ahlfors homotopy (el. the proof of L e m m a 6.10) of w~ into the identity. Using (9.8) the Ahlfors homotopy can be projected into a homotopy of 0]

(U-A)/G=O] (U/G)-

{d} into the[identity. This concludes the argument.

Since L e m m a 6.9 is purely topological, we have actually established it whenever S has a finitely generated fundamental group, since every such surface is homeomorphie to

1 2 4 ~ P M ~ BLURS

a compact R i e m a n n surface with at m o s t finitely m a n y punctures. We can now repeat the proof of L e m m a 6.10 a n d establish Theorem 9 for all finitely generated groups G.

Now let G be infinitely generated. There is a sequence of finitely generated subgroups G 1, G 2 ... of G with

l # G I C G ~ c G a c . . . , G=G1UG2UGaU ....

We are given the universal covering v: U - ~ U - A with covering group V c G . L e t Aj denote the Gj orbit of a, so t h a t

A I C A ~ c A a ~ .... A = A 1 U A 2 U A s U ....

L e t vj: U-+ U - A j be the holomorphic universal covering determined b y the conditions:

vj(i) =v(i), v~(i)v'(i) > 0 .

We can repeat the construction of the m a p p i n g Q given in w 6, cf., in particular, L e m m a 6.5. There are uniquely determined torsion free Fuchsian groups Vj, ~j, conformal bijections uj: U/Gj~(U-A~)[Gj and exact sequences

1-~ Vjc-, ~j ZJ , Gj-+ I, where V~ is the covering group of vj, and

hjo~=zj(~)oh j for ~EG~.

F o r every j there is a bijection

Q~: L~(U, Gj)I ~ L~(U, Gj)I such t h a t v=~j(/~) if and only if

v(vj (z)) vj (z)/v~ (z) = #(z). (9.9)

Observe t h a t L~(U, G)~Loo(U, G t) for all ?'.

LEMMA 9.1. Let vEL~(U, G)I and set/~j=~yl(v) ] = 1 , 2 ... Then

lira w,j(z)=wg(z) /or z E U U R. (9.10)

Proo]. A standard function theoretical a r g u m e n t shows t h a t lim vj (z) = v(z) for z e U

F I B E R SPACES OVER TEICHMULLER SPACES 125 uniformly on compact subsets (cf., for instance, the proof of L e m m a 15 in [10]). B y (9.9) and (6.6) we have t h a t

lim ~uj (z) = #(z), z E U.

J - ~

Since also ll ,ll =ll ll <1, the conclusion (9.10) follows (cf., for instance, [a]).

Now we can prove Theorem 9 for G. L e t #, fiELoo(U, 0)1 and assume t h a t v=~(~u) a n d ~=~(/2) satisfy [wv]=[w~], w~(a)=w;(a). We m u s t show t h a t [w~]=[w~]. Define

~ut =O~-l(v)/2j =~71(~) where Oj is the m a p p i n g defined above. Since G s is finitely generated we know t h a t [ w j = [wb.], for all ~. The desired conclusion follows from L e m m a 9.1.

w 10. Teichmiiller spaces ot low genus

L e t M be a complex manifold and, for every mEM, let A(m) be a J o r d a n domain in C. We say t h a t A(m) depends holomorphicaUy on m of there is a continuous m a p p i n g (m, t)~-~Zm(t ) of M • R into C such that, for every fixed t, Zm(t) depends holomorphically on m and, for every fixed m, t~->Zm(t) is an orientation preserving homeomorphism of R onto the b o u n d a r y of A(m). (We orient the b o u n d a r y of a J o r d a n domain so t h a t the domain is to the left, and we consider R as the b o u n d a r y of U.)

Example. L e t G be a Fuchsian group with dim T(G)< 0% Then the J o r d a n domains Du([w~]) and Db([W~]), cf. w 2, depend holomorphically on T = [w~] E T(G). I n the first case, we can set Z~(t)=w~(t), in the second, Z~(t)= W~(t).

A domain A c C r will be called a Bergman domain if either r = 0, or r > 0 and there is a B e r g m a n domain M c U -1 and, for every mEM, a J o r d a n domain A ( m ) c ~ depending holomorphically on m, such t h a t A consists of all pairs (m, z) with m E M and zEA(m).

THEOREM 11. The Teichmi~ller spaces T(p, n) with p=O, n>~3, with p = l , n>~l, and with p =2, n >~0 can be represented as bounded Bergman domains.

Proo/. L e t G be a torsion free Fuchsian group such t h a t T(G) can be identified with T(p, n). L e t ~ / b e related to G as in Theorem 9, so t h a t T(~) can be identified with T(p, n+ 1). Since Fu(G ) and Fb(G) are isomorphic to T(G), we conclude from the example above t h a t T(p, n + l ) is a (bounded) B e r g m a n domain if T(p, n) is.

Now T(0, 3) is a point a n d T(1, 1) is a J o r d a n domain. Hence the s t a t e m e n t of the theorem is true for p = 0 and p = 1. I t is also true for p = 2 in view of the known

L E ~ M A 10.1. T(0, 4) is isomorphic to T(1, 1), T(1, 2) to T(0, 5) and T(0, 6) to T(2, 0).

We sketch a proof, for the sake of completeness. I f G is a Fuchsian group with

1 2 6 LrPMAN BERS

[16]. TEICHM~LLER, O., Ver/~nderliche Riemannsche F1/iehen. Deutsche Mathematilc, 7 (1949), 344-359.

Recived June 30, 1972

Dans le document F(p, X(p, n) Y(p, (p, n) D('v)/G ~. T(p, n) T(p, n), z~D(~). T(p, TM. T(p, n) X(p, n) T(p, n) T(p, n), X(p, n)= T(p, T(p, n) T(p, n) (p, n), Columbia University, New York, N.Y., USA (1) To Elisha Netanyahu FIBER SPACES OVER TEICHMULLER SPACES (Page 30-38)