Topology in Molecular Biology

216 R. Brooks

2π/3

Fig. 12.4.The fundamental domain forPSL(2,Z)

(1 +i

√

3)/2, respectively. Removing the inverse images of these two points
fromS, we now get a covering

g:S−f−^1

(

i,

1+i

√

3

2

)

→S^2 −{ 3 .points}

The points we remove fromSare clearly removable singularities ofg,soby
filing them in, we get a branched cover ̃g:S→S^2 branched only over three
points.
It is easy to see that (1) is equivalent to (2). Indeed, the oriented graph
(G,O) exhibitsSOas an orbifold covering ofH^2 /P SL(2,Z). To see that (1)
implies (3), we may lift the horocycle onF joiningitoi+1 toSO.The
corresponding system of horocycles has the desired properties. Conversely,
suppose thatSOhas the desired system of horocycles. Joining two horocycles
ofSOby geodesics if the corresponding horocycles are tangent, we obtain a
decomposition ofSOinto ideal triangles. Each ideal triangle has three horo-
cycles perpendicular to the edges and pairwise tangent. The only way this can
happen is if the points of tangency are at the pointsi, i+ 1, and (i+1)/2. It
follows thatSO=SO(G,O) for some (G,O).
We will now show:

Theorem 4 (Belyi).LetSbe a closed Riemann surface. Suppose there exists
anumberfieldKand a holomorphic function

f:S→S^2

whose critical values lie inK∪{∞}.