8
H. BLAIN E LAWSON , JR.
where x = x + ix an d a C satisfie s \a\ 1. Eac h suc h transformatio n map s th e
boundary circl e ont o itself . Recal l tha t thi s grou p i s isomorphi c t o P5L
2
(R) = SL2(R)/Z
It i s eas y t o chec k tha t th e abov e trivializatio n
(1.7) T^D^^D 2 xS l
is equivarian t wit h respec t t o th e Mobiu s transformations , wher e th e actio n o n T AD 2)
is th e natura l induce d actio n o n tangen t vector s an d th e actio n o n th e righ t i s th e join t
action o n eac h facto r separately . I n fact, thi s actio n i s transitiv e an d ha s trivia l iso -
tropy subgroups ; hence , ther e i s a natura l diffeomorphis m T AD ) ^ PSL
2
(R).
Observe tha t th e trivializatio n (1.7) give s ris e t o a codimension-1, C 0* foliation o f
T AD 2) b y projectio n ont o S . Eac h lea f o f thi s foliatio n consist s o f a field o f uni t
tangent vector s whos e geodesi c ray s ar e asymptoti c (i.e. , hav e th e sam e limi t poin t o n
3D ) . I n particular, eac h lea f i s a unio n o f orbit s o f th e geodesi c flo w o n T AD2). Fur
thermore, i t i s clea r tha t thi s foliatio n i s invarian t b y th e grou p o f isometrie s discusse
above.
We now recal l tha t an y compac t (Riemann ) surfac e o f genu s 1 ca n b e realize d a
a quotien t 2 = D /T wher e T i s a discret e subgrou p o f PSLAK). Hence , th e uni t tan
gent bundl e t o 2 ha s th e for m Tj(S ) = T
x
(D2)/r. Sinc e th e foliatio n o f T
X
(D2) i s in -
variant unde r T , i t descend s .to a codimension-1, C 03 foliatio n o f T,(2) . Thes e folia -
tions ar e calle d Anosov foliations. The y ar e transversa l t o th e fiber s o f th e bundl e
Tj(S)— 2, an d the y ar e quit e complicated .
Analogous foliation s exis t fo r an y complet e Riemannia n manifol d wit h sectiona l
curvature c 0. However , excep t fo r th e constan t curvatur e case , th e differentiabilit
is quit e low . (I t i s C i n general , an d C i f di m M = 2 o r i f th e curvatur e i s !4-pinche
[HP].)
There i s a secon d wa y o f viewin g thi s foliatio n whic h i s usefu l computationally .
Recall tha t T
X
{D) ^ P5L
2
(R). Th e Li e algebr a o f P5L
2
(R) i s generate d b y thre e ele
ments:
-CD
-(:: -(::)
where:
[A, L + ] = L + ; [A, L'] = L" ; [L + , L~] = 2A.
Hence, th e right-invarian t vecto r field s o n PSLJR ) correspondin g t o A an d L spa n
an involutiv e plan e fiel d whic h integrate s t o giv e th e Anaso v foliatio n describe d abov
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