FIBER SPACES OVER TEICHMULLER SPACES
BY LIPMAN BERS
Columbia University, New York, N.Y., USA (1) To Elisha Netanyahu
I n t r o d u c t i o n
We first summarize the results of this p a p e r for the simplest and most i m p o r t a n t special case: the Teichmiiller spaces T(p, n) of surfaces of t y p e (p, n), i.e. closed R i e m a n n surfaces of genus p with n punctures. The points of T(p, n) are equivalence classes of orientation preserving homeomorphisms of a fixed surface S 0 of t y p e (p, n) onto other such surfaces; two m a p p i n g s , / 1 a n d / 2 , are considered equivalent if there is a conformal m a p p i n g h such t h a t / ~ 1 o h o / 1 is homotopic to the identity. H o m o t o p y classes of orientation preserving automorphisms of S o form the modular group Mod (p, n) which acts naturally on T(p, n), and X(p, n)= T(p, n)/Mod (p, n) is the space of moduli (conformal equivalence classes) of surfaces of t y p e (p, n). We assume throughout t h a t 3 p - 3 + n ~>0. The space T(p, n) has a canonical structure of a complex (3p 3 +n)-dimensional manifold, the action of Mod (p, n) on T(p, n) is holomorphic and properly discontinuous, and X(p, n) is a normal complex space.
A central result in Teichmiiller space theory asserts t h a t T(p, n) admits an essentially canonical representation as a bounded domain in C T M . I n proving this result [7] one attaches to every ~ e T(p, n) a J o r d a n domain D@) and a quasi-Fuchsian group G~, b o t h depending holomorphically on ~, such t h a t ~ is the equivalence class of mappings of S0 onto D('v)/G ~. The fiber space F(p, n) over T(p, n) is the set of pairs (~, z), With r E T(p, n), z~D(~).
We shall show tha~ the group Mod (p, n) can be extended to a group m o d (p, n) which acts holomorphically and properly discontinuously on /~(p, n). The quotient Y(p, n ) = F(p, n)/mod (p, n) is a normal complex space and a fiber space over X(p, n) with the
(1) Work partially supported by the National Science Foundation.
9 0 LIPMAN BLURS
property: the fiber over a point x E X(p, n) representing a R i e m a n n surface S is isomorphic to S/Aut (S) where Aut (S) is the group of all conformal automorphisms of S. F o r n = 0 , t h e existence of such a fiber space was asserted, without proof, b y Teichmiiller ~16] and proved, in a completely different way, b y Baily [5].
We shall also establish an isomorphism between T(p, n ~ - l ) and F(p, n) which con- jugates m o d (p, n) into a subgroup of index n § 1 of Mod (p, n § This implies t h a t X(p, n § is a ramified ( n + l ) - s h e e t e d covering space of Y(p, n).
I n the last section the isomorphism theorem is used to represent the space T(p, n) for p = 0 , 1, 2 as B e r g m a n domains, t h a t is as sets of r-tuples of complex numbers (z 1 ... z~) determined b y t h e requirement: zj lies in a J o r d a n domain depending holomorphieally on the variables z~, k <?'. I t is hoped t h a t this representation will prove useful.
I n the b o d y of the p a p e r we deal with a more general case, Teichmtiller spaces T(G) and modular groups Mod (G) of a r b i t r a r y Fuchsian groups G. (If G does not have a funda- mental domain of finite non-Euclidean area, T(G) is a domain in an infinitely dimensional complex Banach space a n d Mod (G) need not act discontinuously.) The construction of the fiber space 2'(G) and of the extended modular group m o d (G) goes through in all cases. The isomorphism t h e o r e m can be stated and proved whenever G has no elements of finite order. We give two proofs of this theorem, one relies on a topological result of D. B. Epstein [12], the other is self-contained.
The results of this p a p e r have been announced without proof in the survey article [9]; this article also contains all needed definitions and results and an extensive biblio- graphy.
I a m grateful to I. K r a , D. B. Patterson and C. J. Earle for reading and criticizing an earlier version of this paper, and to P. Shalen for drawing m y attention to Epstein's paper.
w 1. Teichmiiller spaces
I n this section we fix our notations and recall some basic definitions and facts.
L e t U denote the upper hall-plane of the complex plane; we denote b y Q the group of all quasicon/ormal automorphisms of U, and, for every ~ e Q , we denote b y K(oJ) the dilatation of co. The elements co EQ with K(co) = 1 form the subgroup Qoon~ of eonformal auto- morphisms of U; it can be identified with the real MSbius group. I t is known t h a t every eoEQ can be extended, b y continuity, to an automorphism of the closure of U in the extended complex plane C = (~ U ( ~ }; b y abuse of language, we shall denote this extension b y the same letter co. The elements wEQ normalized b y the conditions w(O)=O, w(1)=l, w(c~) = ~ f o r m t h e subgroup Qnom. Clearly,
F I B E R SPACES OVER TEICHMULLER SPACES 91 Q = QnomQ~ = QoonfQnom, Q~ n Q~o~ = 1,
where 1 denotes the trivial group 1 = {id}.
The set Qno~m is m a d e into a complete metric space (but not into a topological group) b y defining the Teichmiiller distance ~ between two elements w and ~ as
O(w,
d~)
= log K(~bow-1).B y a Euchsian group we mean, in this paper, a discrete subgroup G of Qconf. The region of discontinuity ~(G) of G, i.e. the largest open subset of C on which G acts properly discontinuously, is either the union of U and the lower halfplane L or a domain containing UUL; the set A ( G ) = C - ~ ( G ) is called the limit set of G and is the set of accumulation points of orbits of G. The group G is called of the first or of the second kind according to whether A(G) coincides with the extended real axis R = R U { c~ } or not. For instance, the trivial group 1 is of the second kind.
L e t G be a Fuchsian group (this notation will be kept throughout this paper). A qua- siconformal automorphism eo EQ is called compatible with G if ~oGo-l=Qconf. I f so, (oG(o -1 is a Fuchsian group and is of the first kind if a n d only if G is, a n d the m a p p i n g g~->o~ogoo~ -1 is a (type preserving) isomorphism. The set of elements co EQ compatible with G will be denoted b y Q(G); thus Q=Q(1). We set Qnorm(G)=Q(G)N Qnorm-
Two elements, co a n d eg, of Q will be called equivalent if eolR=~5[R. The elements equivalent to the identity form a normal subgroup Q0c Q~om. I f co and ~5 are compatible with a Fuchsian group G of the first kind, t h e n t h e y are equivalent if and only if the isomorphisms g~->egogoa) -1 and g~+d~ogod) -1 coincide. The equivalence class of w EQ will be denoted b y [co].
The Teichmi~ller space T(G) of a Fuehsian group G is the set of equivalence classes [w] of elements ~o EQnorm(G). The canonical surjection Q(G)~--> T(G) defines a Teichmiiller distance function ~a on T(G) and makes T(G) into a complete metric space. I n particular, the universal Teichmi~ller space T(1) is the factor group Qnom/Qo; it is not, however, a topological group.
I t is clear t h a t if G and G I = G are two Fuchsian groups, t h e n T(G)= T(G1). I t turns out t h a t T(G) is closed in T(G1) and the embedding T(G)r T(G1) is a homeomorphism.
Also, (Sa~ ] T(G) • T(G) ~< (~a; it is not known whether the equality sign holds in general. I n particular, every Teichmiiller space T(G) is a closed subset of the universal Teichmiiller space T(1).
L e t L~o(U) be the usual complex Banach space of (equivalence classes of) b o u n d e d measurable functions, let Loo(U)l be the open unit ball in L~(U), and, for a Fuchsian
92 LteMAN BERS
group G, let Loo(U, G) be the closed linear subspace of L~(U) consisting of elements ~u satisfying
~(g(z)) g'(z)/g'(z)
=/~(z) for g E G.Also, let Loo(U, G)I =L~o(U)I A L~(U, G).
E v e r y ~o EQ has a.e. partial derivatives and a Beltrami coe]]icient ~ = (~o/(~)/(~w/Oz) which belongs to Lo~(U)I; also
I+IIA
= _ I I A
where II II is the Zoo norm. E v e r y /~ELcc(U)I is the Beltrami coefficient of a unique normalized automorphism w EQnorm; we write
W ~W/~
and note t h a t w~EQ(G) if and only if #EL| G).
Thus there is a canonical bijection Loo(U , G)l->Qnorm(G), which is a homeomorphism, and there is a continuous surjection ;u~->[w~] of Loo(U, G) 1 onto T(G) which defines in T(G) a complex structure (of a ringed space). This is the same structure as the one given b y the embedding T ( G ) c T(1), for L(U) has a continuous projection on Loo(U, G).
The Teichmiiller space T(G), with its complex structure, can be realized, canonically, as a bounded domain of a complex Banach space B2(L, G) defined as follows: the elements of B2(L, G) are holomorphic functions ~(z) defined in the lower halfplane L, satisfying the functional equation of quadratic differentials:
q~(g(z))g'(z) 2 =~(z) for gEG, and having a finite norm
II~]]B = s u p y2[~(z)l (z=x+iyEL).
I t is clear t h a t B2(L , G) is a closed linear subspace of B2(L, G1) for every Fuchsian group G 1 c G, and it is known t h a t dim B2(L , G) < ~ if and only if G is finitely generated and of the first kind.
For e v e r y / ~ EL~(U)I there is a unique quasiconformal automorphism w of ~ with w(0) = 0 , w(1)=1, w ( c ~ ) = ~ , such t h a t w I U has the Beltrami coefficient # and w l L is conformal. We write
W : W / ~
and denote b y cf~(z) the Sehwarzian derivative of w~(z) in L : d d (w ,IL).
q~" = u' - 89 2 where u(z) = dzz log dzz
F I B E R SPACES OVER T E I C H M U L L E R SPACES 93 B y N e h a r i ' s t h e o r e m [14], 9~seB2(L, 1) a n d
I1 11 <
~. i t h a s b e e n shown t h a t wslL, w s l I t a n d ~ s d e p e n d only on [ws], t h a t ~ s ~B2(L, G) if ludLow(U, G), a n d t h a t the mapping [ws]~->~s is a biholomorphic bi]ection o/ T(G) onto a holomorphieally convex domain in B2(L, G) containing t h e open ball of radius 89 F r o m n o w on we shall i d e n t i f y T(G) with its canonical i m a g e in B2(L, G).w 2. Fiber spaces over Teichmiiller spaces
T o e v e r y p o i n t v of t h e universal Teichmtiller space T(1) t h e r e is associated a n unbounded Jordan domain Du(~ ) defined as follows. L e t v = [w~]; since w~(L) d e p e n d s o n l y on [w~] a n d n o t on t h e p a r t i c u l a r choice of /4 so does w~(U), t h e c o m p l e m e n t of t h e closure o f y ~ ( L ) . W e set Du(~)-w~(U). T h e b o u n d a r y of Du(v) is t h e directed J o r d a n curve w~(R); it a d m i t s t h e p a r a m e t r i c r e p r e s e n t a t i o n ~ = w~(x), - co <~ x < c~. F o r e v e r y fixed x E It, ~ depends h o l o m o r p h i c a l l y on # EL~o(U), as follows f r o m t h e results of [3], a n d since w~(x) d e p e n d s o n l y on v - [ w ~ ] , ~ is, for a fixed x, a holomorphic function of T E T(G). I n this sense D~(v) depends holomorphically on ~:.
W e also define a bounded Jordan domain Db (~), b y t h e following construction. L e t
~l(Z) a n d ~2(z) be t w o linearly i n d e p e n d e n t solutions of t h e o r d i n a r y differential e q u a t i o n
2U"(z )+~t'(z)~(z) - 0 , z E L (2.1)
n o r m a l i z e d b y t h e initial conditions
U1 ( - i) - ~/~ ( - i) - 1, ~i ( - i) - U2 ( - i) - 0. (2.2)
and set W~(z) - Ul(z)/U~(z). (2.3)
Then, as is well-known, W~ has t h e Schwarzian d e r i v a t i v e ~ , so t h a t t h e r e is a c o m p l e x MSbius t r a n s f o r m a t i o n fl~ such t h a t
W~ ~ / ~ o w ~ (2.4)
in L; we use this relation to define W~(z) for all z E C, a n d we sev Db(~) ~ W~(U). T h e defini- tion is legitimate since W~(U) is t h e c o m p l e m e n t of t h e closure of W~(L) a n d W~IL depends only on [w~],
W e show n o w t h a t fit depends o n l y o n [w~], a n d t h a t this dependence is holomorphic.
N e a r z - - i we h a v e t h a t w~(z)=a+b(z+i)+c(zz~i)2§ where a 4 : 0 , b 4 0 ; here. a , b, c d e p e n d h o l o m o r p h i c a l l y o n / 4 a n d d e p e n d only on [w~], since wz IL d e p e n d s o n l y , o n [ % ] . On t h e o t h e r hand, b y (2.1), (2.2) a n d (2.3) we have, n e a r z = - i , t h a t W ~ ( z ) = ( z + i ) - l + b(z + i) + ~(z + i) 2 + . . . . H e n c e fl~(t) = [ct § ( b 2 - - ac)]/(bt - abi. This o r o v e s t h e assertion. W e conclude, a s before t h a t D0(Y) depends holomorphicalty on ~.
94 LIP~AN BERS
Now let G be a Fuehsian group. The
unbounded fiber space flu (G)
overT(G)
is the set of pairs (v, z) with vET(G),ZEDu(v);
t h ebounded fiber space F~(G)
is the set of pairs (v, z) withveT(G),
zeDb(v); the terminology will be justified below. Both fiber spaces are subsets ofB~(L,
G)(~(5 and restrictions toT(G) of
theuniversal fiber spaces
Fu(1), F~(1).There is a canonical bijection Fu(G)-->i~b(G ) which takes a point
(v,z)EJF~(G),
with v = [w~], into the point (7,fit(z)) of Fb(G ).
T H E O R E ~ 1.
The fiber spaces Fo(G) and _F~(G) are domains, the canonical bijection Fu(G)~ Fb(G) is biholomorphic, and Fo(G) is bounded.
Proo/.
F o r every # ELco(U)I, the function[
z ( ~ ) = - 2 i w . , Ir
is holomorphie and univalent, and Z(0)~ 0, Z ' ( 0 ) - 1 . B y K o e b e ' s one-quarter theorem, I Z(~)] ~> ~ for ]~] = 1. Therefore I
W'a(x) l
~ 2 for x E R. HenceD~([wz])
= W~(U) lies in the disc Iz[ ~<2. Thus Fo(1) is a bounded set inB~(L,
1)|To show t h a t
F~(G)
is a domain it is enough to show t h a t it is open, and this requires to demonstrate the following. !~or every,a o EL~(U,
G1) 1 and every z 0 E Do ([w~.]) there are positive numbers 81 and e 2 such that, for everyvELoo(U , G)
with Ilvll = 1 , a n d for every tEC with it[ <e~, the discIz-zo[
<e~ lies inDb([W~o+tV]).
L e t e~>0 be so small t h a t the disc Iz - %] < 2e, lies in D0([wr176 I t is enough to find an el such that, for all v and t as above, and for all x e R , one hasIW~~
<ca. We choose an e > 0 so small t h a t for v as above and for It] < e,/~0 + tv e L~( U, G h. F o r every x E R the holomorphic function of t, W~~ (x) - W~, (x), vanishes for t = 0 and has a modulus not exceeding 4 for I tl < e. B y Schwarz' l e m m a the desired inequality holds with e 1=ee2/4.
The other assertion of t h e theorem follows from t h e fact, established above, t h a t fl, in equation (2.4) depends holomorphieally on [wz] E
T(G).
Often there is no need to distinguish between the two isomorphic fiber spaces, and one writes simply
F(G).
w 3. The universal modular group
E v e r y wEQ induces an a u t o m o r p h i s m co. of Qnorm defined as follows: if
wEQnorm,
co. (w) is the unique element of Q,om which can be written asco.(w)
= ~owoco -1 (~EQoo,f). (3.1)Note t h a t g depends on w. Each co. is an isometry and a holomorphic m a p p i n g of Qnom,
F I B E R SPACES OVER TEICHlYIULLER SPACES 95 and since [o).(w)] depends only on [~o] and on [w], co. m a y be considered as an (isometric and holomorphic) automorphism [~o], of T(1), which depends only on [col and maps [w] ~ T(1) into [~. (w)]. The group of all these automorphisms is called the universal modular group and is denoted b y Mod (1); it can be identified with the factor group Q/Qo. I t turns out t h a t the action of Mod (1) on T(1) can be extended to an action on the fiber space F(1) over T(1) which respects the fiber space structure.
THEOREM 2. The group Mod (1) operates on Fu(1), as a group o/ holomorphic auto- morphisms, according to the rule: i/ o~ E Q, w = w ~ /or some # EL~o( U)I, and z E Du([W ~]) = w~(U), then
[o~], ([w~], z)---- ([w~], 2) (3.2)
where ~e L ~ ( U)I with co, (w,) = w~ (3.3)
and ~ = w~o ~oo (w~) -1 (z). (3.4)
(The action of Mod (1) on F~(1) is defined similarly.)
The proof is somewhat long and will be broken up into several lemmas.
L E M ~ A 3.1. T h e mapping z ~ - ~ de/ined by (3.4) is a con/ormal bijection o/ Du([w~]) onto Du([W~]) and depends only on the equivalence classes [w~], and [o)] (and not on the particular choices o//~ and co).
Proo]. Let h~ be defined b y
w ~ = h~ow~ I U; (3.5)
then h~: Du([w~])-* U is a eonformal bijection since w~ and w~ I U have the same Beltrami coefficient/~. This bijection keeps 0, 1, co fixed, hence it depends only on [w~]. Similarly, w, = h vow" I U, and the conformal bijection h~: Du([w~])--> U depends only on [wv] = [ o . (w~)]
and thus only on [w~] and [~o]. Now, b y (3.4) and (3.3),
= h~-i owvo coow~loh~ : h - l o (~ow~ow-1)oogow~loh~ = h~ lo ~oh~.
Since ~Qoo.~, and ~ clearly depends only on [wv] and [eo], the assertion follows.
L e m m a 3.1 implies t h a t the right side of (3.2) depends only on [o)], [w~] and z.
LEMMA 3.2. For # and z as in Theorem 3, a n d / o r ~Ol, o~EQ, we have [ ~ o o ~ ] . ([w~], z) = [~l].o[~].([w~], z).
The proof is b y a direct verification and is left to the reader, The lemma implies t h a t Mod (1) is a group of bijections of Fu(1).
L E P T A 3.3. Every [o~], is a continuous sell-mapping o/ F~(1).
96 L I P M A N B E R S
Proo/. Since it is k n o w n t h a t t h e restriction of [w], on t h e "first coordinate" [w~]
is an isometric self-mapping of T(1), it suffices to p r o v e t h e following. L e t w be fixed and let #, #~, j = 1, 2 ... be elements in Lo0(U)l such t h a t lira 81([wzj], [wz]) = 0. F o r e v e r y j, a n d for z ew~ (U), set s = wVieoJe (w~) -1 (z), where w,j = w.(w~). T h e n lira ~j= 3, uniformly on compact subsets of w~ (U).
Since s depends only on t h e equivalence class [w~j] we loose no generality in assuming t h a t lim(~(w~,w~)=O; t h e n also lim6(w~j, w , ) = 0 . Using s t a n d a r d properties of quasi- conformal mappings one verifies t h a t w~, (w~) -1 and w~ converge, uniformly on c o m p a c t sets, to w~, (w~) -1 a n d w ~, respectively. W h e n c e t h e assertion.
LWMMA 3.4. Let t~-->a~ be a holomorphic mapping o/the disc It I <e in C into Loo(U)~.
There is an s o, 0 < % < e , and a holomorphic mapping te->#t o/ Itl <c0 into Lo~(U)I such that (i) fit(z), z e U is, /or ]t I <e0, a real analytic /unction o/ x = R e z , y = I m z , and, /or every compact set A ~ U and every integer n > 0 , the moduli o/tzt and its partial x and y derivatives up to the order n are bounded on A by a constant depending only on A and n, and (if) [/tt] = [~t] /or It/< Co.
Pro@ T h e r e are constants 00 and 0 such t h a t 0 < 0 0 < 1 , 0 < 0 < 8 9 and if T E L l ( U ) and II~ll
<0o,
t h e n [l~0~l[B<0 (here II II is the Loo norm, I[ lib t h e n o r m defined in w 1, and~v ~ t h e Sehwarzian derivative of w~[L).
W r i t e the given w~0 in the form w~o=we, o.:.ow~, where qjeL~o(U)t and ![eJ[[ <00, j = l , ...,r. This is easily done; Observe, for instance, t h a t for e v e r y aELoo(U) with [[a[[ < k < ] , we have t h a t wa=w~ow, m where [[v[[ ~<kl[[a[[ with kl, 0 < k l < l, depending only on k.
N e x t , set T j ( z ) = - 2 y 2 ~ ( 5 ) . T h e n ~ e L ~ ( U h, [[~[[ <0, and b y the Ahl/ors-Weill lemma (cf. [4]), ~*~ = ~e~, so t h a t [w~i ] -- [we~]. We define/~0 b y the condition w~0 = w,~o.., o w,r T h e n [wz,] = [w,,].
N o w let e0, 0 < e o < e be so small that, for I t [ < e0, if we define ~t ~Loo(U)~ b y the require- ment: w,~ = w~,owo-~ ~, t h e n H~tH < oo so t h a t [IV "~ II < 0. Set ~t (z) = - 2y2~v "~ (~), and d e f i n e / h b y the requirement: wgt= wr T h e n the mapping t~->~t is holomorphic and so are the mappings t~-->$t and t~+#t. Also, [wet/= [wn~ ], b y the Ahlfors-Weill lemma~ so t h a t [ % J = [ % o w,.] = [w, o w J = [ % ] .
Thus t~-->~t satisfies condition (if). N o t i n g t h a t t h e n u m b e r s
IlvOql
... II~0ql andII ,ll
are a~l b o u n d e d b y the same constant 0< 89 one verifies t h a t condition (i) is also satisfied.L ~ a 3.5. For a [ixed ~o~Q and /or a /ixed (relevant) z, the number ~ de[ined by (3.4) depends holomorphicaUy on tt ~Loo( U)r
F I B E R S P A C E S O V E R T E I C H M U L L E R S P A C E S 9 7
Proo/. I t is required to prove t h e following. L e t t ~-->#t be a holomorphic m a p p i n g of t h e disc [ t ] < s in (~ into L~o(U)I. L e t z 0Ew ~~ a n d l e t e0, 0 < s 0 < s , be so small t h a t z 0 E w € (U) for ] t I < e0" L e t w~ t = co, (w,t) and, for z E w "t(U), set
= gt(z) = w ~t o ~ o (w'O -1 (z). (3.6) T h e n t~+gt(zo) is a holomorphic function of t n e a r t = 0 .
I n view of L e m m a s 3.1 a n d 3.4 we m a y assume t h a t t~--~/xt has p r o p e r t y (i) of L e m m a 3.4. The Beltrami e q u a t i o n ~w"t/~2=#t(z ) (~w~t/~z) is an elliptic s y s t e m of two first order partial differential equations for t h e real a n d i m a g i n a r y p a r t s of w ~. This s y s t e m is u n i f o r m l y elliptic as long as ]]#t[[ is b o u n d e d a w a y f r o m 1. U n d e r this hypothesis t h e solutions w € (z) are u n i f o r m l y b o u n d e d on c o m p a c t sets in U. Using s t a n d a r d t h e o r y of elliptic partial differential equations we conclude f r o m p r o p e r t y (i) that, for [t]<s0, w't(z) is real analytic in x a n d y a n d t h e partial derivatives of w ~t (z), up to a n y given order, are u n i f o r m l y b o u n d e d on c o m p a c t subsets of U. I t is k n o w n (cf. [3]) that, for a n y fixed z E U, t h e n u m b e r w "t (z) depends holomorphically on fit a n d hence on t. I n view of the r e m a r k m a d e above, the same is t r u e of t h e partial derivatives of w ' t (z). F o r instance,
~ w " ( z ) _ lim w ' ~ ( z + h ) - w~(z)
~x R~h~O h
is a holomorphic function of t, since t h e limit is a t t a i n e d u n i f o r m l y in t.
Next, t h e Beltrami coefficient vt of w, (w~t) depends holomorphically on fit (as is k n o w n a n d easy to check) a n d hence on t. Therefore w ~t (z) is, for e v e r y fixed z e U, a holo- morphic function of t; it is u n i f o r m l y b o u n d e d for It] < s 0 a n d z restricted t o a c o m p a c t set in U.
t~inally, let ~o= (w'~ let r be a n u m b e r with 0 < r < I m $0, a n d let Ct be t h e image u n d e r w't of t h e circle ~ = ~0+ re ~~ 0 ~< 0~<2~. T h e n Ct is a s m o o t h J o r d a n curve, a n d if It] is small enough, z 0 lies in t h e d o m a i n interior to Ct. W e restrict ourselves to such t a n d a p p l y C a u e h y ' s f o r m u l a t o t h e holomorphic function zc-->gt(z), cf. L e m m a 3.1.
N o t i n g (3.6) we h a v e t h a t
i
1 _f~ gt (z) dz
_ _ 1 (2~ g t o w ~ t ( ~ o + re'~ ~w~t(~o+ rei')
dO 1 (e~ w~t o co(~ o + re ~~ Ow"e ($o + re~~ dO
- - ~ , ~ - - ~ o - - -
27ei Jo w (~o+re ) - z o ~0 This f o r m u l a exhibits ~gi(z0) as a holomorphic function of t.
7 - - 732904. A c t a mathematica. 130. I m p r i m 6 le 1 F 6 v r i e r 1973.
98 LrPMAN BERS
This lemma implies t h a t the mapping [co].: Fu(1)-~Fu(1) is holomorphic. Theorem 2 is now proved. The verification of the following theorem is trivial.
THEOREM 3. Let G be a Fuchsian group and let wEQ(G). Then [w],[E(G) is an isomorphism o/ F(G) onto F(o~Gw-1).
Here _~ m a y be interpreted as either _F u or F b. B y an isomorphism we mean a biholo- morphic bijection which respects the fiber space structure. I n particular [w], maps T(G) holomorphically onto T(eoGeo-1). I t is easy to check t h a t this latter mapping is a Teichmiiller isometry in the following sense: the ~ distance is t a k e n into the 6a~
distance, GI=wGo -1.
The mappings [co], IT(G) and [co], IF(G), for oJeQ(G), are called allowable iso- morphisms.
w 4. Modular groups
Let N(G) and ~Yco~ (G) be the normalizers of G in Q and in Qconf, respectively. The extended modular group rood (G) of a Fuchsian group G is defined as the subgroup of the universal modular group Mod (1) induced b y N(G); thus rood (G) can be identified with the quotient N(G)/(N(G)~ Qo). In view of Theorem 3 the elements of rood (G)induce allowable automorphisms of Fu(G), and also of F~(G).
L e t gE_hrconf(G). Then the coset [~] contains no elements of Qconf distinct from ~.
We m a y therefore, b y abuse of language, identify [~] and [a], with ~. Hence G m a y be considered as a subgroup of rood (G), a normal subgroup, of course. The modular group Mod (G) of G is defined as the factor group
Mod (G) = rood (G)/G ~- (N(G)/(N(G) fl Qo))/G (4.1) The element of Mod (G) induced b y o~EN(G) will be denoted b y (oJ).
One verifies easily that, for g e G a n d w E Qnom(G), one has g, (w) = w. Hence [g], I T(G) = id, and, b y Theorem 3, every element (co) of Mod (G)induces an allowable automorphism [r ) of T(G). However, the action of Mod (G) on T(G) need not be effective; a non-neutral element of Mod (G) m a y induce the identity mapping on T(G).
THEOREM 5. Let wEQ(G) and ~=eoGeo -1. The allowable isomorphism T(G)-~T(~), F ( G ) ~ P(~) conjugate the actions o/Mod (G) and o / r o o d (G) into those o/ Mod (G) and o/
rood (~).
The proof is clear.
T H e O R e M 6. The action o/ Mod (G) on T(G) is e//eetive i/ G is o/ the first kind and its
F I B E R S P A C E S O V E R T E I C H M U L L E R S P A C E S 99 signature is not (0, 3; ~1, v~, va) with at least two o/the ~j equal, (l, l; u), (1, 2, ~, v) or (2, 0).
The action of m o d (G) on F(G) is always effective.
Before p r o v i n g this t h e o r e m we recall h o w one defines t h e signature (p, n; ~1, ~2 ... ~n)
of a finitely g e n e r a t e d F u c h s i a n g r o u p of t h e first kind. T h e n u m b e r p is t h e genus of U/G, n is t h e n u m b e r of n o n - c o n j u g a t e m a x i m a l elliptic or parabolic subgroups of G, a n d vl .... , vn are t h e orders of these subgroups, a r r a n g e d in ascending order. I f G has no torsion, t h e o n l y possible value for a uj is oo a n d t h e signature is d e t e r m i n e d b y t h e type
(p, n)
of G. W e recall t h a t d i m T ( G ) = 3 p - 3 + n . T h e o n l y restrictions on t h e signature of a F u e h s i a n g r o u p are: p>~0, n>~0, 2 ~ < v j < ~ a n d 2 p - 2 + n - ( 1 / u ) - . . . - ( 1 / v n ) > O .
Proo/ o/ Theorem 5. Assume t h a t t h e action of Mod (G) on T(G) is n o t effective. This m e a n s t h a t there is an r ) such t h a t
[:r 1] = [w] for all weQnor~(G), ~eQconf depending on w, (4.2)
[eo0] g=[g] for all geG. (4.3)
A p p l y i n g (4.2) to w = i d we o b t a i n t h a t [ a o o -1] = [id]. H e n c e we m a y assume t h a t co 0 = a, so t h a t
a) o e N~on~(G) - G. (4.4)
One sees b y (4.4) t h a t eoo induces a eonformal self-mapping 0 =kid of U/G such t h a t t h e d i a g r a m
O) o
U , U
~]/o 101 , u/a
(4.5)
is c o m m u t a t i v e . H e r e (and hereafter) u n m a r k e d vertical arrows d e n o t e n a t u r a l projec- tions. One sees f r o m (4.5) t h a t 0 moves a p o i n t over which t h e covering U-~U/G is ramified of order y into a n o t h e r such point. This implies a l r e a d y t h a t if t h e t y p e of G is (0, 3) or (1, 2), two of t h e orders ~ m u s t be equal.
N o w consider some w E T(G), a n d let a be d e t e r m i n e d f r o m (4.2). One checks at once t h a t ~ E N(wGw -1). Also ~r ~ wGw -1, for otherwise t h e r e would be a go E G with ~ = w o go o w -1 a n d (4.2) would i m p l y t h a t [w0] -- [g0], contradicting (4.3). As before, ~ induces a eonformal self-mapping of U/wGw -1 distinct f r o m t h e i d e n t i t y a n d preserving t h e orders of rami-
I00 L r P ~ N B]~S
fication of the covering U--->
U/wGw -1.
B u t if 0 < d i m T(G)< co and the t y p e of G is not (0, 3), (1, 1), (1, 2 ) o r (2, 0), t h e n one can find a wET(G) so t h a t U/wGw -1 admits no such conformal automorphisms. One sees this b y noticing t h a t a "general" compact R i e m a n n surface of genus 1o > 2 , a torus with n > 2 "general" punctures, and a sphere punctured at n > 3 " g e n e r a l " points admit no non-trivial conformal automorphisms.Assume now t h a t (4.2) and (4.3) holds. To complete the proof we m u s t show t h a t for
~u a n d v in L~(U, G)I , connected b y the relation w , = w , (wg), the equality
2 = z (4.6)
cannot hold for all zEwg(U); here ~ is given b y equation (3.4). B u t we have [wg]=[w,]
b y (4.2) and therefore wg=w v. I f (4.6) holds for all zEwg(U), t h e n wg] U=wgoO)o] U or e % = i d which contradicts (4.3).
Remark. Theorem 5 could be strengthened if the following statement were true.
E v e r y R i e m a n n surface which is not the three times punctured sphere, a once punctured torus, or a closed surface of genus 2, and which does not admit a continuous group of conformal automorphisms, is quasieonformally equivalent to a R i e m a n n surface which admits no non-trivial conformal automorphisms whatsoever.
This sounds quite reasonable, b u t I know of no proof.
THEOREM 7. I/ dim T ( G ) < ~ , then the groups Mod (G) and m o d (G) act properly discontinuously on T(G) and F(G), respectively.
Proo]. We m a y assume t h a t dim T(G)> O, otherwise T(G) is a point, F~(G)= U and m o d (G) a Fuchsian group. I t is enough to prove t h a t the groups in question are discrete.
The discreteness of Mod (G) is a classical result of Fricke. Nevertheless we sketch a proof, for the convenience of the reader.
Since dim T(G) < ~ , the group G h a s in U a fundamental domain which is compact except for finitely m a n y "parabolic cusps". Using this one shows easily t h a t for every A > 0 there are only finitely m a n y non-conjugate hyperbolic elements g E G such t h a t the non-Euclidean distance between some point z E U a n d the point g(z) is less t h a n A. This implies t h a t the set
{tit = (trace (g))~, g E G} is discrete. (4.7)
~Let gl, g2, ..., gT b e d s e t of generators f o r G chosen so t h a t gl and g2 are hyperbolic, the fixed p o i n t s of gl being s e p a r a t e d b y those of g2. Such generators exist since dim T(G) > O.
One can find ;a finite s e t : l ~ o f words 71, 72, ..., ?~ in :gl, . . , g r such t h a t the sequence {(trace (?j))~} determines the sequence {gj}, except f o r d conjugation in Qoonf.
I~IBER SPACES OVER TEICHMULLER SPACES 101 Assume n o w t h a t {~o.} c N(G) is such t h a t lim @o~)([wJ)=lim [ w . . ( w ) ] = [ w ] f o r e v e r y w E T ( G ) . I n particular l i m [ ~ % . ( i d ) ] = [ i d ] , so t h a t there a r e elements a.EQconf with a . o w -1 EQ.om a n d lira [a~ow -1] = [id]. Hence, for e v e r y g E G,
lim (trace ( ~ o w~ 1 ogo eo~o ~1))2 = (trace
(g))2.
I n view of (4.7) we have, for n large a n d g ~ F ,
(trace (~o~ 1 oqo o~n)) 2 = (trace (if))2.
Recalling h o w F has been chosen we see t h a t there exist e l e m e n t s / ~ E Qr such t h a t , for n large a n d g E G,
(On I ogor n =/~nOgOfln ] .
This implies t h a t [(o~ 1] = [fl,]. W i t h o u t changing [~o,] we m a y assume t h a t ~o, = fl~l, so t h a t
~o~E2Voon~(G ). Since Noonf(G)/G is k n o w n to be finite (this follows f r o m t h e hypothesis t h a t d i m T ( G ) < r we conclude t h a t (~o~} ([w])= [w] for all w E T(G) a n d all large n. T h u s Mod (G) is discrete.
N o w let ( ~ o , } c N ( G ) be such t h a t
lim [o~].(v, z ) = (v, z) for (v, z)EFu(G). (4.8) L e t ttELoo(U, G)I , let unEL~(U,
G)i
be such t h a t w,~ =w~.(w~), a n d let 2~ be d e t e r m i n e d b y t h e relation~ = w ~ o c o ~ o ( w " ) - l ( z ) , z e w ~ ( U ) , eL equation (3.4). N o w relation (4.8), for v = [w~], reads
lira [w,~] = [w~], lim s = Z. (4.9)
B y t h e previous a r g u m e n t we k n o w t h a t [~%] = [y~], y,~ENr a n d [w,.] =[we] for large n. F o r such n we m a y assume, w i t h o u t changing 2., t h a t w~ = y~, a n d w,~ =w~, so t h a t w ' , = w~, zn = weo ~ o (w~) -l(z). T h u s t h e second e q u a t i o n (4.9) implies t h a t lira yn(z) = z for z E U. Since Nco~ (G) is k n o w n to be a F u c h s i a n g r o u p we conclude t h a t g~ = id, ~ = z for large n.
Thus [w~],(v, z) = (~, z) for large n. This shows t h a t rood (G) is discrete.
w 5. Fiber spaces over m o d u l i spaces
I n this section we consider only finitely g e n e r a t e d F u c h s i a n groups G of t h e first kind.
Given two such groups, G a n d G, t h e existence of a w EQ(G) with coGo~ -1 = G is equivalent to t h e condition t h a t G a n d G h a v e t h e same signature. I f this condition is satisfied, t h e n
102 ~ r ~ M ~ BWRS
there exists a w EQno~(G) such t h a t wGw -1 is c o n j u g a t e to G (in t h e g r o u p Qoo~,). F u r t h e r - more, if w a n d ~ are t w o elements of Qnom(G), t h e n t h e groups wGw -1 a n d ~ G ~ -1 are c o n j u g a t e if a n d o n l y if there is a o~EQ(G) with co.(w)=~o.
I t follows f r o m T h e o r e m 3 t h a t T(G) a n d F(G) are d e t e r m i n e d up to isomorphisms b y t h e signature (p, n; Vl ... ~ ) of G. So are, b y T h e o r e m 5, t h e groups Mod (G) a n d rood (G).
T h e n o t a t i o n s T(p, n; ~1 ... v~), F(p, n; ~1 ... v~), Mod (p, n; Vl, ..., v~), m o d (p, n; vl ... v~) are therefore legitimate. W e m a y also d e n o t e t h e quotients
X(G) = T(G)/Mod (G), Y(G) = F(G)[mod (G)
b y X(p, n; vl . . . ~ ) a n d Y(p, n; v~ . . . v~), respectively. Observe t h a t X(T, n; Vl . . . v~) is t h e space o/moduli (conjugacy classes) of F u c h s i a n groups of signature (p, n; vl ... Vn).
B y T h e o r e m 7, a n d b y a general t h e o r e m of H. C a r t a n [11], X(G) a n d Y(G) are n o r m a l complex spaces. Recalling t h e definition of t h e action of m o d (G) on F(G), we see t h a t there is a n a t u r a l holomorphic surjection Y(G)->X(Y), induced b y t h e m a p p i n g ('~, z)~+z of F(G) o n t o T(G).
L ~ M A 5.1. Let #EL~(U, G)I. The inverse image Z~ o] the point [w~] under the mapping Y(G)~X(G) is isomorphic to the quotient
U/Nconr(Wt~Gw~l).
Proo/. W e i n t e r p r e t Y(G) as Fu(G)/mod (G). I n view of Theorems 3 a n d 5 we loose no generality in assuming t h a t [w~] = [id] or even t h a t ju = 0. W e m u s t therefore consider t h e s u b g r o u p F of m o d (G) which keeps t h e fiber ([id], z), z E U of F~(G) over t h e point [id]E T(G) fixed a n d determine t h e q u o t i e n t Z 0 = Du([id])/F = U/F. Now, an element co E1V(G) induces a n element [co]. E F if a n d o n l y if [w]. ([id])=[id], i.e., if a n d only if there is a n aEQconf such t h a t [no o) -1] = [id], i.e., if a n d only if [ ~ ] . = [a]., a E Noo~f(G). H e n c e ~0 = U/Ncont(G), as asserted.
W e can reformulate t h e result as
T~]~OR~M 8. There is a normal complex fiber space over the space o/moduli o/Fuchsian groups with signature (p, n; vl .... , vn), the ]iber over a point representing the conjugacy class o / a group G being isomorphic to U/Nconf(G).
I f G has no torsion, we write T(p, n) ... Y(p, n) instead of T(p, n; v 1 ... vn) etc.
I t is k n o w n t h a t T(p, n), Mod (p, n) a n d X(T, n) are t h e Teichmtiller space, t h e m o d u l a r g r o u p a n d t h e space of moduli, respectively, of c o m p a c t R i e m a n n surfaces of genus p with n punctures, as defined in t h e introduction. I t is also k n o w n ([10], cf. also [13]) t h a t t h e r e are canonical isomorphisms
T(p, n; v 1 ... ~,~) -~ T(p, n)
FIBER SPACES OVER TEICHMi)7.J~ER SPACES 103 which induce isomorphisms between Mod (p, n; ~1 ... v~) and subgroups of Mod (p, n).
I t follows from classical uniformization theory t h a t there are canonical bijections X ( p , n; ~1 .. . . . ~n) -~ X ( p , n; ~ . . . ~ )
(5.1)
whenever the signatures are such t h a t vj =vJ+l if and only if ~j =~j+l" I t is also easy to see t h a t there are canonical surjections
X ( p , n; ~/1 .. . . . Yn) "-~ X ( p , n).
(5.2)
I t is not difficult to show t h a t (5.1) is biholomorphic a n d (5.2) is a finitely m a n y sheeted ramified holomorphic covering.
We note t h a t Theorem 8 has the following
COROLLARY. There exists a normal complex fiber space Y ( p , n ) over the space X(T, n) o] moduli o] compact Riemann sur/aces o/genus p with n punctures such that the/iber over a point o / X (p, n) representing the con/ormal equivalence class o] a Riemann sur/ace S is isomorphic to S/Aut (S) where A u t (S) is the group o/all con/ormal sell-mappings o / S . For n~-0 this was conjectured b y Teichmiiller [16] and proved, in an entirely different way, b y Baily [5].
w 6. The isomorphism theorem
I n this and the following sections we consider only torsion /ree Fuehsian groups.
We shall show t h a t for such a group G the fiber space F(G) is isomorphic to a Teichmtiller space T(~) for another Fuchsian group G.
THEO~WM 9. Let G be a torsion/tee Fuchsian group, a a point in U, d the image o / a under the natural projection U-~ U/G. Let G be another torsion /ree Fuchsian group and u: U/~ ~ ( U / G ) - (d} a con]ormal bi]ection. Then there is a canonical isomorphism (biholo- morphic bijection) T(G) ~ F(G).
Before proving the theorem we make some r e m a r k s which will be used also in the following section.
Remark 1. We recall t h a t a puncture P of a R i e m a n n surface S is defined b y a domain D c S and a conformal bijection 0 of D onto the unit disc punctured at the origin (the domain 0 < ]~] < 1 ) such t h a t a sequence ( P j } c S , with limO(Pj):O, diverges on S.
A n y such sequence is said to converge to P. There is an obvious equivalence relation and a
104 L I P M A N B E R S
natural way of topologizing the union of S and the set of its punctures, and of making this union into a Riemann surface.
Now assume t h a t S = U/F, F a Fuehsian group. Then there is a natural one-to-one correspondence between the punctures P of S and the Conjugacy classes of maximal parabolic subgroups F 0 of F, defined as follows. Let A be a fundamental region of F0 in U, and let z 0 be the fixed point of the parabolic generator of F0. The subgroup F 0 belongs to P if, given any sequence {z j} c A with lira zj = %, the image of this sequence under the natural projection U ~ U/F converges to P.
Under the hypotheses of Theorem 7, there is a distinguished puncture of UIG which m a y be denoted by u-l(d).
Remark 2. We recall t h a t an open arc of an ideal boundary curve of a Riemann surface S is defined b y a domain D c S and a conformal bijection 0 of D onto the unit disc with the property: a n y sequence {Ps}c D for which $ = l i m 0(Pj) exists, diverges on S if and only if I m ~ > 0, I ~1 = 1; such a sequence is said to converge to a point on an ideal boundary curve. There is an obvious equivalence relation and a natural way of topologizing the union of S and its ideal boundary curves.
Now assume t h a t S = U/F, F a Fuchsian group. Then the ideal b o u n d a r y curves of S can be identified with the components o f
&-A(r))/r,
and the projection V---> U/F extends, b y continuity, to a mapping (UUR)-A(F)-+[(UUtt)-A(F)]/F. Note t h a t F is of the first kind if and only if U/F has no ideal b o u n d a r y curves.Remark 3. Under the hypotheses of Theorem 7, if G and d are given, the point a is determined but for an action of G. We denote the set of all points g(a), g C G, b y A. This set is determined by G and d; it is infinite except if G = 1. The choice of G and d determines G up to a conjugation in the group Qoonf of real MSbius transformations. To find G, choose a holomorphic universal covering v: U-+ U - A and let G consist of all y CQoonf for which v(z) and voy(z) are always G equivalent. Then there is a commutative diagram
U V , U - A
UlO
> ( U - A ) I O = ( U / G ) - { a }(6.1)
where unmarked vertical arrows denote natural projections. Clearly, u is a conformal bijection. There is an exact sequence
1--> V r (~ g , G ~ 1 (6.2)
where V is the covering group of v, and
F I B E R S P A C E S O V E R T E I C H M U L L E R S P A C E S 105
voy---X(y)ov for VE0. (6.3)
Conversely, given G, a, G and u, there is exactly one v satisfying the above conditions.
Indeed, since u can be lifted to a conformal bijection between the universal covering surfaces of U/O and of (U - A)/G, and since the covering U - A -+ (U - A)/G is subordinated to a universal covering, we have a c o m m u t a t i v e diagram
U 0c , u - v o , ( U - A )
~ / 0 ' ( u - A ) / a
(6.4)
where ~EQconf and Vo, v 1 are holomorphie universal coverings. Non, set v = v o o ~ ; then (6.4) becomes (6.1).
I n proving Theorem 7 we shall work with the unbounded fiber space Fu(G). The proof will be given in a sequence of lemmas.
LEM~IA 6.1. I1 G = I , V = G is a cyclic group with a parabolic generator. I / G~=I, V contains in/initely many hyperbolic elements with distinct/ixed points.
Proo/. I f G = I , t h e n U - A = U - { a } . This implies the first statement. I f G 4 : l , the fundamental group of U - A , which is isomorphic to V ~ G is infinitely generated. This implies the second statement.
LEMMA 6.2. The groups G, ~ and V are either all three o/the/irst kind or all three ol the second kind, I n all cases, A ( V ) = A ( 0 ) . I / G is o~ the second kind the diagram (6.1) extends by continuity to the diagram
U U (fi - A(G)) -
[u
u(ft- A(0)]/0 1
U, ( U - A ) U (R- A(G))
, [(V - A) U (R -
1
A(G))J/G(6.5)
(by abuse of language we do not distinguish between u, v and their continuous extensions).
The assertion of the l e m m a follows from R e m a r k 2 above and f r o m the following two observations. The ideal b o u n d a r y curves of ( U / G ) - {d) can be identified with those of U/G, b y means of the m a p p i n g u. The ideal b o u n d a r y curves of U - A are the components of R - A ( G ) , since A(G) is the set of accumulation points of A.
106 L I P M A N B E R S
Lw~MA 6.3. The mapping R: Loo(U, G)I-->E~(G) which sends a vELoo(U, G)I into ([wv], wv(a)) e Fu(G ) is a holomorphic surjection. The complex structure o/Fu(G) could have been de]ined by means o] this surjection.
Proo]. Let v E L ~ ( U , G)I be given and let A be a simply connected fundamental domain for t h e group Gv=w~G(w,) -1 in D~([w~])=wv(U) with a smooth b o u n d a r y and containing w ~ (a) in its interior. One can deform the m a p p i n g w v in the interior of A, and then, correspondingly, in all fundamental regions w~ogo (w~) -1 (A), g E G, which are equiva- lent to A, so t h a t the new m a p p i n g remains quasiconformal and has a Beltrami coefficient a E L~0 (U, G)I with [w,] = [w,], and w~ is equal to a given point in A. Using this r e m a r k
one verifies easily t h a t R is a surjection.
The other statements of the l e m m a follow from the fact t h a t w~(a) is a holomorphic function of vELoo(U, G) 1 and t h a t the differential of JL:c(U, G)2-~ T(G) is, at each point, surjective, and its kernel has a c o m p l e m e n t a r y subspace.
Lw~MA 6.4. The mapping R: L~o(U, G)I-~ T(G ) • U which sends a vELoo(U, G)I into the pair ([w,], wv(a)) is a real analytic sur]ection and there is a real analytic bi]ection,
l: F~(G) -* T(G) • U with 1~ = lo R.
Proo]. We have t h a t wr=h~ow v ] U where h~ is the conformal m a p p i n g of Du([W~])=
w~(U) onto U normalized b y the conditions h~(0)=0, h~(1)=l, h~(oo)= co, cf. the proof of L e m m a 3.i. Since h~ depends only on [w~],
F~ (G)~ ([w,], z) ~ ([w~], h~(z)) e T(~) • V 1
is a bijeetion and l o R = l ~ . The rest at the proof is left to the reader. Note t h a t w~(a) is known to depend real analytically (though not holomorphically) on v ELoo(U, G)I.
L]~MMA 6.5. There is a linear isometric bi]ection ~: Loo(U, G ~ L o o ( U , G) de/ined by the condition: ~ = e(#) q and only i]
~,(v(z)) v'(z)/v'(z) = #(z), (6.6)
or equivalently, i] and only i/ there is a commutative diagram
V w# 9 g
w, = wq(~)
I
U - A , w ~ ( U - A )
(6.7)
where v~ is a holomorphic universal covering. I] so, then the covering group o/ v~ is w~ Vw71.
(By abuse of language we do not distinguish between w v and w~ I U - A.)
F I B E R SPACES OVER T E I C H M U L L E R SPACES 107 Proo]. If ~,ELoo(U), t z in (6.6) is well defined. If ttELoo(U) is given, ~ is also well defined, since if v(~) = v(z), then ~ = y(z) for some ~ E V, and hence v'(z) = v'(?(z))7'(z) so t h a t
~(~)
v '(~) / v' (~) = ~(r(z)) v' (~,(z) ) / v' (~,(z) )= ~(r(z)) [v'(z)/v'(z)]r'(z)/r'(z) = ~(z) v'(z)/v'(z).
Let veLoo(U, a), and assume t h a t (6.6) holds. We have, b y (6.2) and (6.3), t h a t to every 7 E O there is a g E G with v(r(z)) = g(v(z)), v'(7(z))?'(z ) = g'(v(z))v'(z). Hence, b y (6.6),
#(~(z)) 7'(z)/7'(z) = ~(g(v(z) ) )g' (v(z) ) v' (z) / g' (v(z) ) v' (z) = ~,(v(z) ) v' (z) / v' (z) = tt (z), so t h a t / ~ e L ~ ( U ,
O).
One computes similarly t h a t if (6.6) holds, andfieLd(U, 0),
then 9 ~Lo~(U, G). I t is clear t h a t ~ is linear and norm preserving.For a given ;u ELo~(U, 0)1, define ~ b y (6.6) and lift the mapping w~ [ U - A to universal covering surfaces. This yields a commutative diagram
U O) , U
U - A , w , ( U - A )
1
(6.s)
where ~ is a holomorphic universal covering. Here we m a y replace ~ by ~o ~ and r b y ~-1 ow, where ~ EQco~f. One computes t h a t co is quasiconformal and t h a t r =tt. Hence there is a unique ~ such t h a t o~-lo~o=w~. With this ~, set v~=~oo~; then (6.8) becomes (6.7).
The existence of the commutative diagram (6.7), with a holomorphic v~ implies relation (6.6); the proof is a calculation.
From (6.7) we see t h a t a ?EQ~o~f satisfies v~o?=v~ if and only if WvOVOW~Io~2 =
w~ovow~ 1, t h a t is, if and only if v o w ~ l o ? o w ~ = v . This is so if and only if w71o?ow~E V.
Hence w~ Vw71 is the covering group of v~.
LEMMA 6.6. Let tto and ttl be elements o/ L ~ ( U , G)I with [w,0]=[w~,], and set
~'o=~(/~o), ~'1=~(/~1). ~hen -~(~'o)---~(~1)"
Proo/. Since [W~o]=[w~~ W~oOyOW~,l~wmoyow[~ for all ~E0. In particular, the groups W~oVW~, 1 and w~,Vw~f coincide, so t h a t W ~ o ( U - A ) and w ~ ( U - A ) are con- formally equivalent (cf. the preceding lemma). We have therefore the commutative diagram
U w]~~ ~ U L w#~ U
( U - A ) w~. , W~,o(U-A) c~ , w ~ ( U - A ) , w ~ ( U - A )
1 0 8 LIPMAN BERS
where ~ is a conformal bijection. Since A is discrete in U, so are w~, (A) and w~, (A), a n d we conclude, b y the theorem on removable singularities, t h a t ~ is (the restriction of) an element of Qoo~f.
I t follows from L e m m a 6.2 t h a t if G is of the second kind the above d i a g r a m extends b y continuity to the following
UU ( t t - A ( ~ ) ) w z o UU (R-A(wz0~w;.~)) w # , U 0 ( R - A ( 0 ) )
vl 1 v
U# w~~ , w~.(U#) ~ ) w ~ , ( U # ) 9 w~ , U #
(6.10)
where U s = (UU t t ) - ( A U A(G)).
B y hypothesis, w~ 1 owz0 commutes with all elements of G. B y (6.9) and (6.3) it follows t h a t
w~lo~OW~o
commutes with all elements of G, hence leaves all fixed points of hyper- bolic elements of G fixed, hence leaves all points of A(G) fixed. B y hypothesis,w ~ o w , o
leaves every point of R fixed, and it follows from (6.10) t h a tw~loo:ow~o
leaves every point of R - A ( G ) fixed. Thusw;-.~oo~ow~oltt=id.
Since w% and w~, leave 0, 1, fixed so does ~. Hence ~ = i d and we conclude t h a t [w,0]=[w~,].Observe now t h a t the point a m a y be considered as a puncture on
U - A
=v(U), and let P 0 be a m a x i m a l parabolic subgroup of V belonging to this puncture (cf. R e m a r k 1 above). L e t Yo be a generator of F 0. To simplify writing, assume t h a tyo(z)=z+l
(this can be achieved conjugating G in Qconf), L e t A denote the region 0 ~ Re z < 1, I m z > 0.I f {zj} is a sequence in A with lira [zj[ = o~, then
limv(zj)=a.
Set A'=w;~owv0(A).Since
w;,low~,o
is an automorphism of U U R which commutes with alI elements of G, A' is also a fundamental region for P0 in U. For every sequence {zj}c A' with lira ] z ' [ = oo, we have t h a t lirav(z;)=a.
Noting t h a t ~ = i d , we obtain from (6.9), for n = l , 2 ....v o w , i ow,o (in) = w;J OW~oOV(in).
For n-~ ~ this yields a = w~]o w~0 (a) on w~l ( a ) = w~~ (a). This completes the proof of L e m m a 6.6.
L ~ A 6.7.
There is a holomorphic surjection T(G)-+ Fu(G ) which,/or every # ELo~(U, ~) takes
[w~]into
R(Q(/~)).I/this surjeetion is injective, it is biholomorphic.
This follows at once from L e m m a s 6.3, 6.4, 6.5, and 6.6.
We
now
prove Theorem 7 for the special case G = 1.~ I B E R SPACES O V E ~ TEICHMT.ILLER SPACES 109 L E ~ M A 6.8. I] G = I the sur]ection T(G)~Fu(G ) is injective.
Proo/. To simplify writing we assume (without ]oss of generality, cf. L e m m a 6.1) t h a t a = i a n d t h a t v: U ~ U - { i } is chosen as
9 1 § e 2~z
v(z) =~-1 - e ~ " (6.11)
L e t #EL~(U, G)I a n d v=~(~u). T h e n v~: U - ~ U - ( w , ( i ) } m u s t be of t h e f o r m v~=~ov with ~EQoo.f a n d ~(i)=w~(i). T h e group V = G is g e n e r a t e d b y z~-~z+l a n d w~ satisfies w~(z+l)=w~(z)+c, with some c > 0 . Setting z = 0 we see t h a t c = l . W e use t h e con- tinuous extension of t h e c o m m u t a t i v e d i a g r a m (6.7), n o t i n g t h a t A ( G ) = {c~ }. F o r 0 < x < 1 we o b t a i n t h a t w , ( - c o t ~ x ) = a ( - c o t T ~ w ~ ( x ) ) . L e t t i n g x-~0 we o b t a i n ~ ( ~ ) = ~ , so t h a t
a(z) = v(z) I m w,(i) § Re wv(i). (6.12) F o r O < x < l we o b t a i n f r o m (6.7), (6.11) a n d (6.12) t h a t
w,( - cot ~x) = - (cot 7ewe(x)) I m w,(i) + R e w~(i).
T h u s t h e knowledge of /~(v)=([wv], w,(i)) determines w~(x) for O < x < l a n d hence for all x E R .
T h e rest of t h e proof of T h e o r e m 9 will be based on a topological t h e o r e m b y D. B.
E p s t e i n which asserts t h a t a n a u t o m o r p h i s m of an orientable surface with base point which is h o m o t o p i c t o t h e i d e n t i t y is isotopic to t h e i d e n t i t y ([12], p. 101). A c t u a l l y we need only a weak corollary of this result which we state as
L E ~ M A 6.9. Let S be a Riemann sur/ace which is not homeomorphic to the sphere, the plane, the punctured plane or to a torus. Let d E S and let 0 be a topological orientation- preserving automorphism o/ S with O(d)=d. Assume that 0 is homotopic to the identity by a
homotopy which leaves d /ixed. Then O I S - ( d ~ is homotopic to the identity9
I n w167 8 a n d 9 we shall give a proof of T h e o r e m 9 which does n o t assume L e m m a 6.99 L E ~ M A 69 The surjection T(~)-->Fu(G ) is injective.
Proo/. W e assume t h a t G ~:1, otherwise t h e r e is n o t h i n g to prove (cf. L e m m a s 6.1 a n d 6.8)9 Since G :~1, we loose no generality in assuming t h a t 0, 1, ~ are a m o n g t h e fixed points of elements of ~; this can be achieved b y conjugating ~ in Qr (cf. L e m m a 69
~ o w let ju 0 a n d ~ul be t w o elements of L~o(U, G)I a n d set vo =~(juo), v1=~(#1). Assume that/~(v0)
=-]:~(~)1)"
W e m u s t show t h a t [w,,] = [w,1 ].110 L r P ~ N B~RS
Our hypothesis implies t h a t
W, oOgOw~l=w~,ogow~, ~
for allgEG,
so t h a t the two groups,w,~ 1
and w~,Gw~ 1, are the same; we denote this group b y Gv. The hypothesis also implies t h a tw,~ w,,(a), W,o(A)=
w,,(A); we denote this point and this set b y a~and Ap, respectively. Finally, the hypothesis implies t h a t
w,,ow~ 1
commutes with all elements of Or. L e t d r denote the image of a~ under the projectionU ~ U]G~.
Now we construct the
Ahl]ors homotopy d)~
betweenw,~ t
and the identity: for every t, 0~<t~<l, and for e v e r yz6U, d~(z)
is that point on the non-Euclidean segment joiningw,.ow~l(z)
to z which divides the non-Euclidean length of this segment in the ratiot/(1 -t).
Then cSt(z) depends continuously on (t, z) and, for everygEG, r
Hence there is, for every t, a commutative diagram
which, for t = O, becomes
~o t
U , U
1 1
U/G, cot , U/G,
U w ' o ~ : U
U/O, coo , U/G,
(6.13)
(6.14)
where co o is a homeomorphism onto. For t = 1 we have col =id, &l =id. Now egt is a homotopy of
w~~ 1
into the identity keeping a~ fixed, so t h a t cot is a homotopy of coo into the identity keeping d r fixed. B y L e m m a 6.9 there is a homotopy ~ t ofcool(U]Gv)-{av}
into the identity. Using L e m m a 6.5 (cf. diagram (6.7)) and restricting diagram (6.13) we construct the commutative diagram
U , w,, U w~ , U
V,ul
* w,,I U - A w,.[ U - A
U - A ~ , U - A , U - A ,
l COo,(U_A)[G " 1
( U - A,)/G, , ( U - A,)]G,
(6.15)
Using the outer square of this diagram we lift the homotopy ~t and obtain, for each t the commutative diagram
FIBER SPACES OVER TEICHMOLLER SPACES 111
U ~ , U
v/x, l Iv/x~
U - A~ U - A,
( u - A , ) I O , > ( U - A , ) l a ,
(6.16)
where ~ t (z) depends continuously on (t, z), ~o = W/xo O w~/, ~o = we
[( u -
Av)/Gv, ~ t = id, and :r denotes the natural projection.The mappings ~rov,~ and ~rov/x 1 are universal coverings, and v/x,, v/x~ are universal coverings of the same domain U - A~. Hence there is an ~ EQconf with v/x, = v,~ I t follows easily from L e m m a 6.5 and from (6.1), (6.2), (6.3) t h a t the covering groups of ~rov/x~ and
~rov~, are ~ o = w,~ 1 and ~ = w/x, ~ w ; / , respectively. We conclude from (6.16) t h a t for every ~ e O 1, every t and ever z e U, there is a unique ~ e 0 0 such that ~toZ(z)=~ofi~(z).
Since this ~ must depend continuously on z and on t, and 00 is discrete, ~ depends only on ~. Since ~ l = i d we have ~OV/XoO5 t =;T~OV/Xt =;71:OVIX00~. Hence 510~--le00 and 51eQoon f.
Now, for every 7E~1, we have w/x~ 1. Hence
w/x~ B y hypothesis, A(~) contains the points 0, 1, oo; so does A(dl) since A(G1)=w/x,(h(d)). B u t w~~ ~ leaves 0, 1, co fixed. So does therefore ~ t , and since ~ is a Mhbius transformation, ~ 1 = id. Hence
Since ~x = id, w,~. V w ; 1 = w., V w ; / . Of V~o.
w/x.ow~/]A(G0)
=id. (6.17)we have t h a t ~ - - 1 ~ O, Hence v/x,=V/xoOlT---:v/x o. Also, Go=(~l and This latter group will be denoted b y V~; it is the covering group The upper half of the commutative diagram (6.15) extends, b y continuity, to
(U U It) - A(G0)
V/X~
[( u u ~l) - (A u A(a,)) w~~
, (U U R ) - A((~0)
l V/xo
, (U o t i ) - (A U A(G,))
(by abuse of language we use the old names for the various mappings involved). For x in some component I of R - A(G0) we have v/x0 o w/x~ o w~ l(x) -- w~~ o W~l I O V/x o (X) = V/X, (X). Hence there is a (unique) element ~ E V~ with w/x~ = ~(x). Since ~ depends continuously on x and V~ is discrete, ~ is the same for all x in I. The endpoints of the interval I belong
112 LIPMAN BERS
to A(d0); we conclude from (6.17) that y leaves these endpoints fixed, so t h a t y ( I ) = I . Let U denote the maximal subgroup of Vg which leaves I fixed, so t h a t ~EF. Since P is
!somorphic to the fundamental group of vs. (1), a component of 1~ - A(G,), and all compo- nents of R - A ( G v ) are homeomorphic to R, F = I . Thus y = i d and we conclude that
w, oow21It - A(0o) = id. (6.18)
Together with (6.17) this shows t h a t w,, [ tt = w,, [ R.
Thus L e m m a 6.10 is proved, and so is Theorem 9. We note the COROLLARY. There is a canonical isomorphism T(p, n + l )-> F(p, n ).
w 7. A relation b e t w e e n m o d u l a r groups
We recall t h a t b y Theorem 6 the extended modular group rood (G) acts effectively on F(G). B y the same theorem, and under the hypotheses of Theorem 9, the modular group Mod (G) acts effectively on T((~) provided di m T(G) < ~ and G is not of type (1.1). Indeed, U/G cannot be an (unpunctured) closed surface of genus 2 (since it has at least one puncture) or a once punctured torus, or a thrice p u n c t u r e d sphere (since U/G can be neither an unpunctured torus nor a twice punctured sphere).
T~V, OR~M 10. Under the hypotheses o/ Theorem 9, the isomorphism T(G)-+E(G) induces an isomorphism between (the action o]) a subgroup Mod 0 (G) o/ Mod (G) and the group rood (G).
I / U/G is compact, then Mod 0 (G)=Mod (G). I / U/G has precisely n punctures, the index o / M o d 0 (G) in Mod (G) is n § 1.
Note. t h a t U/G has precisely n punctures if and only if G has precisely n conjugacy classes of maximal parabolic subgroups.
Before proving the theorem we note two immediate consequences.
COROLLAnY 1. The isomorphism T(p, n + l ) ~ F(p, n) induces an isomorphism between rood (p, n) and a subgroup o] index n + l o~ Mod (p, n + l ) .
COROLLARY 2. There is a holomorphic surjection Y(p, n) -~ X(p, n § 1)
which is an isomorphism, /or n = 0 , an (n + l)-sheeted fatal/led,covering/or n > 0 .
