Comment. Math. Helvetici 60 (1985) 480-495

0010-2571/85/030480-16501.50 + 0.20/0 ~) 1985 Birkh~iuser Verlag, Basel

H o m o m o r p h i s m s o | Fuchsian groups to PSL (2, ~ ) MARK J~qr,aNs a n d W~a_x~R N

~

r

Introduction L e t F = F ( g :o~1..... Olin) b e a c o c o m p a c t F u c h s i a n g r o u p of genus g with b r a n c h indices c~ . . . . . o~,. In this p a p e r w e d e t e r m i n e the set of c o m p o n e n t s of H o r n (F, P S L (2, ~)). This was d o n e by t h e first a u t h o r in [J] for g > 0 so we o n l y p r o v e the genus z e r o case. T h e m a i n results a r e in t e r m s of an euler class e : H o m (F, P S L (2, ~)) --, H2(F; 7/) d e f i n e d as follows: e(f) = f*(c) w h e r e f* : H2(B P S L (2, ~), )7) --~ H2(BF; )7) = H 2 ( F ; 7/) is the m a p i n d u c e d by f a n d c ~ H2(B P S L (2, ~), Z) ~)7 is a g e n e r a t o r . In [JN] it is s h o w n t h a t H2(F,)7) is t h e a b e l i a n g r o u p : H 2 ( F ; 7/) = ab(xo . . . . . x, I alxi = x0; i = 1 . . . . .

n),

(this c a n easily be s e e n by c o n s t r u c t i n g an explicit BE: r e p l a c e each of n discs in a s u r f a c e of g e n u s g b y a B(7//cti)). T h u s a n y x ~ H2(F, 7/) has a u n i q u e r e p r e s e n t a tion

x=bxo+I3~xl+ . . . +[3,x,,,

0---/3i 0 then 2 - 2 g - n - < b - 2 g - 2 ; (ii) / f g = O then either (a) 2 - n < - b < - - - 2

1Research partially supported by the NSF. 480

Homomorphisms of Fuchsian groups to PSL (2, R)

481

or

(b) b = - 1 and Y4"--1(iS.,]a,) 0.

Proof. We first consider the case [tr AI < 2. Since the set of B e SL (2, R) with tr B = tr A has two components, we only need show that these components are distinguished by sgn ( c - b). Thus we consider

a+d=t (Itl 2 and by t r A together with sgn ( c - b) if Itr AI- 0. Thus

tr A = tr

((ac

b)(cosO

d ,,sin O

-sine~)=2costOcosO_(c_b)sinO. cos O / /

By e q u a t i o n (2) in the p r o o f of L e m m a 1.2 we see that c - b (= 2w in the notation of that proof) lies in the interval [2~/(1 - t2/4), o~) with t = 2 cos to. It is easy to see that c - b takes on all these values. Therefore, since x / ( 1 - t2/4)= 2 sin tO, we see that tr A takes on all values in the interval ( - ~ , 2 cos ( 0 + tO)]. This completes the p r o o f of case a(i); the o t h e r cases are similar calculations. C O R O L L A R Y 2.3. The following regions are S = S(sh (3'1) . . . . . sh (7.)) for the indicated 7i with 0 < 7i < 1 and n > 1.

(a) 0< ~ ~,~ 1 c o m m u t a t o r s is .

w

.

.

.

.

~'.

....

t~, ",:.

x : ~ ~.~:~."

..~

Proofs of Theorems 1 and 2

W e use the notation of the introduction. T h e case g > 0 is d o n e in [ E H N ] (it also follows easily with o u r present approach); we thus assume now g = 0. F = F(0; a l . . . . . (ql . . . .

,

or,) has presentation

q, I qT'= 1, m "

" q, = 1).

H o m o m o r p h i s m s of Fuchsian groups to PSL (2, R)

489

G i v e n a h o m o m o r p h i s m f : F - - ~ G = P S L (2, R), an e q u i v a l e n t w a y to define t h e e u l e r class e(f) is as follows. T a k e t h e p u l l - b a c k c e n t r a l e x t e n s i o n f r o m F

If

0 --~ 7] --~ 0 ---~ G -'~ 1. This gives a d i a g r a m 0 ---~2[ ---./~ ---. F ----~1 (1) O~i~-*

0 ~ G--* 1

w h e r e /" has a p r e s e n t a t i o n (by lifting q~ c F to an e l e m e n t , which we also call q . in/") (ql . . . . .

q., z [ qT' = z a', ql " " " q,, = z-b, Z central)

with 0-O, ad-bc=l and c - b + sd = T. With the change of coordinates used in the p r o o f of 1.2 these become: the

conditions

W - y2

Z2 = 1

t2

4'

(t = 2 cos 0)

w>O

2w + s ( t - 2y)= T. The first two conditions determine one sheet of a hyperboloid of two sheets. T h e third e q u a t i o n describes a plane which thus intersects this sheet in a c o n n e c t e d set.

w

Transverse foliations in Seifert fibrations

Let M be a Seifert fibered 3-manifold with normal f o r m Seffert invariant (g; b; (al, 131). . . . . (a,, 13,)). In [ E H N ] it was suggested that the numerical conditions of T h e o r e m 1 might be necessary and sufficient for M to admit a c o d i m e n sion 1 foliation transverse to its fibers; this was p r o v e d for g > 0 but only sufficiency of the conditions was p r o v e d for g = 0. H e r e we give an example to show that the conditions are not necessary for g = 0. Recall that 9 § = {g :N ~ N I g m o n o t o n i c and g(r + 1) = g(r) + 1 for all r e N}. Let P = F ( g ; b; (12,1, ~31) . . . . . (~ fin)) be the central extension of F = F ( g ; a l . . . . . a . ) defined in Section 3 with center generated by z. In [ E H N ] it is shown that M has a transverse foliation if and only if there exists a h o m o m o r p h ism f : F - - + 9 + with f ( z ) = s h ( 1 ) . (This is equivalent to the existence of a h o m o m o r p h i s m f : F - + H o m e o + ( S 1) with euler class e(f)=bxo+[31x1+" "+

13.x..) By our results we can find R 1 , R 2 , R 3 E G with Ri - sh (4) for i = 1, 2, 3 and

R t R 2 R 3 = s h (2). L e t Qi = p21RiP2 for i = 1, 2, 3 where P2:~---~ R is m a p P2(r) = 2r. Clearly, Qi ~ 9 + and Q i - s h (~) for each i. M o r e o v e r Q1Q2Q3=sh (1). This gives a h o m o m o r p h i s m / : / ~ ( 0 ; - 1 ; (5, 2), (5, 2), (5, 2)) ~ 9 + with [(z) = sh (1),

492

MARK JANKINS AND WALTER NELIMANN

showing the existence of a transverse foliation on M ( 0 ; - 1 ; (5, 2), (5, 2), (5, 2)). This is the claimed counterexample. Analogous constructions using Pm :[~ --* R, P,,(r) = mr give many more exampies. We have strong evidence that if M(0; b; (al,/31) . . . . . (a,,/3,)) admits a transverse foliation, then a transverse foliation can be constructed by this method; details will appear elsewhere (see [JN2]).

Appendix. A homeomorphism for Fig. 1 In the above paper we need only such topological information related to Fig. 1 as is easily available by inspection, so we did not give an explicit homeomorphism. This appendix supplies such a homeomorphism, with a sketch proof. P R O P O S I T I O N . Let D denote the open disk of radius 89and parametrize R x D as • x D = {(z, r, 0) [ z ~ R, 0 sh(z)'( tan(Trr)(c~

0

-cosSin0( ~ ] r+sec r)(~0]

01))-"

But

(rcosO

tan (Trr)/r, sec (wr), and \ r sin 0

rsinO0)

- r cos

are analytic, so ~ is analytic. Using this description, the jacobian of W at any point with r = 0 is also easily seen to be nonsingular, so analyticity of q0 follows by the inverse function theorem.

BIBLIOGRAPHY [EHN] D. EISENBUD,U. HIRSCH,and W. D. NEUMANN,Transverse foliations of Sei[ert bundles and self homeomorphism of the circle. Comment. Math. Helvetici 56 (1981), 638-660. [Ga] MARIADEECAaMENGAZOLAZ,Fibr~.sde Seifert: classification et existence de feuiUemges. C.R. Acad. Sc. Paris 295 (1982), 677-679. [G] W. GOLDMAN,Topological componentS of representation spaces of surface groups. In preparation.

Homomorphisms of Fuchsian groups to PSL (2, •) [J] [JN] [JN2] [M]

495

M. JANKINS, The space of homomorphisms of a Fuchsian group to PSL (2, R), Dissertation (Maryland 1983). M. JANKINS and W. D. NEUMANN, Lectures on Seifert manifolds, Brandeis Lecture Notes 2 (March 1983). M. JANKINS and W. D. NEUMANN, Rotation numbers of products of circle homeomorphisms, Mat. Annalen (to appear). J. MILNOR, On the existence of a connection with curvature zero, Comment. Math. Helvetici 32 (1957-58), 215-223.

College o[ Charleston, Charleston, SC29401, USA University of Maryland, College Park, MD 20742, USA Received July 3, 1984

0010-2571/85/030480-16501.50 + 0.20/0 ~) 1985 Birkh~iuser Verlag, Basel

H o m o m o r p h i s m s o | Fuchsian groups to PSL (2, ~ ) MARK J~qr,aNs a n d W~a_x~R N

~

r

Introduction L e t F = F ( g :o~1..... Olin) b e a c o c o m p a c t F u c h s i a n g r o u p of genus g with b r a n c h indices c~ . . . . . o~,. In this p a p e r w e d e t e r m i n e the set of c o m p o n e n t s of H o r n (F, P S L (2, ~)). This was d o n e by t h e first a u t h o r in [J] for g > 0 so we o n l y p r o v e the genus z e r o case. T h e m a i n results a r e in t e r m s of an euler class e : H o m (F, P S L (2, ~)) --, H2(F; 7/) d e f i n e d as follows: e(f) = f*(c) w h e r e f* : H2(B P S L (2, ~), )7) --~ H2(BF; )7) = H 2 ( F ; 7/) is the m a p i n d u c e d by f a n d c ~ H2(B P S L (2, ~), Z) ~)7 is a g e n e r a t o r . In [JN] it is s h o w n t h a t H2(F,)7) is t h e a b e l i a n g r o u p : H 2 ( F ; 7/) = ab(xo . . . . . x, I alxi = x0; i = 1 . . . . .

n),

(this c a n easily be s e e n by c o n s t r u c t i n g an explicit BE: r e p l a c e each of n discs in a s u r f a c e of g e n u s g b y a B(7//cti)). T h u s a n y x ~ H2(F, 7/) has a u n i q u e r e p r e s e n t a tion

x=bxo+I3~xl+ . . . +[3,x,,,

0---/3i 0 then 2 - 2 g - n - < b - 2 g - 2 ; (ii) / f g = O then either (a) 2 - n < - b < - - - 2

1Research partially supported by the NSF. 480

Homomorphisms of Fuchsian groups to PSL (2, R)

481

or

(b) b = - 1 and Y4"--1(iS.,]a,) 0.

Proof. We first consider the case [tr AI < 2. Since the set of B e SL (2, R) with tr B = tr A has two components, we only need show that these components are distinguished by sgn ( c - b). Thus we consider

a+d=t (Itl 2 and by t r A together with sgn ( c - b) if Itr AI- 0. Thus

tr A = tr

((ac

b)(cosO

d ,,sin O

-sine~)=2costOcosO_(c_b)sinO. cos O / /

By e q u a t i o n (2) in the p r o o f of L e m m a 1.2 we see that c - b (= 2w in the notation of that proof) lies in the interval [2~/(1 - t2/4), o~) with t = 2 cos to. It is easy to see that c - b takes on all these values. Therefore, since x / ( 1 - t2/4)= 2 sin tO, we see that tr A takes on all values in the interval ( - ~ , 2 cos ( 0 + tO)]. This completes the p r o o f of case a(i); the o t h e r cases are similar calculations. C O R O L L A R Y 2.3. The following regions are S = S(sh (3'1) . . . . . sh (7.)) for the indicated 7i with 0 < 7i < 1 and n > 1.

(a) 0< ~ ~,~ 1 c o m m u t a t o r s is .

w

.

.

.

.

~'.

....

t~, ",:.

x : ~ ~.~:~."

..~

Proofs of Theorems 1 and 2

W e use the notation of the introduction. T h e case g > 0 is d o n e in [ E H N ] (it also follows easily with o u r present approach); we thus assume now g = 0. F = F(0; a l . . . . . (ql . . . .

,

or,) has presentation

q, I qT'= 1, m "

" q, = 1).

H o m o m o r p h i s m s of Fuchsian groups to PSL (2, R)

489

G i v e n a h o m o m o r p h i s m f : F - - ~ G = P S L (2, R), an e q u i v a l e n t w a y to define t h e e u l e r class e(f) is as follows. T a k e t h e p u l l - b a c k c e n t r a l e x t e n s i o n f r o m F

If

0 --~ 7] --~ 0 ---~ G -'~ 1. This gives a d i a g r a m 0 ---~2[ ---./~ ---. F ----~1 (1) O~i~-*

0 ~ G--* 1

w h e r e /" has a p r e s e n t a t i o n (by lifting q~ c F to an e l e m e n t , which we also call q . in/") (ql . . . . .

q., z [ qT' = z a', ql " " " q,, = z-b, Z central)

with 0-O, ad-bc=l and c - b + sd = T. With the change of coordinates used in the p r o o f of 1.2 these become: the

conditions

W - y2

Z2 = 1

t2

4'

(t = 2 cos 0)

w>O

2w + s ( t - 2y)= T. The first two conditions determine one sheet of a hyperboloid of two sheets. T h e third e q u a t i o n describes a plane which thus intersects this sheet in a c o n n e c t e d set.

w

Transverse foliations in Seifert fibrations

Let M be a Seifert fibered 3-manifold with normal f o r m Seffert invariant (g; b; (al, 131). . . . . (a,, 13,)). In [ E H N ] it was suggested that the numerical conditions of T h e o r e m 1 might be necessary and sufficient for M to admit a c o d i m e n sion 1 foliation transverse to its fibers; this was p r o v e d for g > 0 but only sufficiency of the conditions was p r o v e d for g = 0. H e r e we give an example to show that the conditions are not necessary for g = 0. Recall that 9 § = {g :N ~ N I g m o n o t o n i c and g(r + 1) = g(r) + 1 for all r e N}. Let P = F ( g ; b; (12,1, ~31) . . . . . (~ fin)) be the central extension of F = F ( g ; a l . . . . . a . ) defined in Section 3 with center generated by z. In [ E H N ] it is shown that M has a transverse foliation if and only if there exists a h o m o m o r p h ism f : F - - + 9 + with f ( z ) = s h ( 1 ) . (This is equivalent to the existence of a h o m o m o r p h i s m f : F - + H o m e o + ( S 1) with euler class e(f)=bxo+[31x1+" "+

13.x..) By our results we can find R 1 , R 2 , R 3 E G with Ri - sh (4) for i = 1, 2, 3 and

R t R 2 R 3 = s h (2). L e t Qi = p21RiP2 for i = 1, 2, 3 where P2:~---~ R is m a p P2(r) = 2r. Clearly, Qi ~ 9 + and Q i - s h (~) for each i. M o r e o v e r Q1Q2Q3=sh (1). This gives a h o m o m o r p h i s m / : / ~ ( 0 ; - 1 ; (5, 2), (5, 2), (5, 2)) ~ 9 + with [(z) = sh (1),

492

MARK JANKINS AND WALTER NELIMANN

showing the existence of a transverse foliation on M ( 0 ; - 1 ; (5, 2), (5, 2), (5, 2)). This is the claimed counterexample. Analogous constructions using Pm :[~ --* R, P,,(r) = mr give many more exampies. We have strong evidence that if M(0; b; (al,/31) . . . . . (a,,/3,)) admits a transverse foliation, then a transverse foliation can be constructed by this method; details will appear elsewhere (see [JN2]).

Appendix. A homeomorphism for Fig. 1 In the above paper we need only such topological information related to Fig. 1 as is easily available by inspection, so we did not give an explicit homeomorphism. This appendix supplies such a homeomorphism, with a sketch proof. P R O P O S I T I O N . Let D denote the open disk of radius 89and parametrize R x D as • x D = {(z, r, 0) [ z ~ R, 0 sh(z)'( tan(Trr)(c~

0

-cosSin0( ~ ] r+sec r)(~0]

01))-"

But

(rcosO

tan (Trr)/r, sec (wr), and \ r sin 0

rsinO0)

- r cos

are analytic, so ~ is analytic. Using this description, the jacobian of W at any point with r = 0 is also easily seen to be nonsingular, so analyticity of q0 follows by the inverse function theorem.

BIBLIOGRAPHY [EHN] D. EISENBUD,U. HIRSCH,and W. D. NEUMANN,Transverse foliations of Sei[ert bundles and self homeomorphism of the circle. Comment. Math. Helvetici 56 (1981), 638-660. [Ga] MARIADEECAaMENGAZOLAZ,Fibr~.sde Seifert: classification et existence de feuiUemges. C.R. Acad. Sc. Paris 295 (1982), 677-679. [G] W. GOLDMAN,Topological componentS of representation spaces of surface groups. In preparation.

Homomorphisms of Fuchsian groups to PSL (2, •) [J] [JN] [JN2] [M]

495

M. JANKINS, The space of homomorphisms of a Fuchsian group to PSL (2, R), Dissertation (Maryland 1983). M. JANKINS and W. D. NEUMANN, Lectures on Seifert manifolds, Brandeis Lecture Notes 2 (March 1983). M. JANKINS and W. D. NEUMANN, Rotation numbers of products of circle homeomorphisms, Mat. Annalen (to appear). J. MILNOR, On the existence of a connection with curvature zero, Comment. Math. Helvetici 32 (1957-58), 215-223.

College o[ Charleston, Charleston, SC29401, USA University of Maryland, College Park, MD 20742, USA Received July 3, 1984