Topology in Molecular Biology

12 The Spectral Geometry of Riemann Surfaces 215

After a M ̈obius transformation, we may assume that the three points are 0,
1, and∞.Ifweset
SO=S−f−^1 { 0 , 1 ,∞},

we may characterize Belyi surfaces in the following way:

Theorem 3.Sis a Belyi surface if and only if there is a finite set of points
{z 1 ,...,zn}onSsuch that any one of the following conditions is fulfilled on
SO=S−{z 1 ,...,zn}:

1.SO=H^2 /Γ,Γa finite index subgroup ofPSL(2,Z).

2. There is a graphGand an orientationOonGsuch that

SO=SO(G,O).

3.SO carries a horocycle packing – that is, a system of closed horocycles

{Ci}about the cusps{zi}such that theCi’s have disjoint interiors, and

the region exterior to all the horocycles consist of triangular regions.

We note that neither the Belyi functionfnor the oriented graph (G,O)
are determined byS.Iffis such a function, then composingfwithz→zn
produces another Belyi function, which gives rise to a new graph (G,O). Proof
of Theorem 3:
Let us first show that (1) holds if and only ifSis a Belyi surface. IfSis
Belyi, thenS−f−^1 ({ 0 , 1 ,∞}) is a covering space ofS^2 −{ 0 , 1 ,∞}.But

S^2 −{ 0 , 1 ,∞}=H/Γ 2 ,

where

Γ 2 =

{(

ab

cd

)}

≡±

(

10

01

)

(mod 2)

Conversely, suppose that there is a finite set{z 1 ,...,z 2 }inSsuch that
S−{z 1 ,...,z 2 }=H^2 /Γ for some Γ⊂PSL(2,Z).
We recall the well-known fundamental domainF 0 forPSL(2,Z) acting on
H^2 actingH^2 – it is traditionally written as

F 0 ={z∈H:− 1 / 2 ≤R(z)≤ 1 / 2 ,|z|≥ 1 }

but we will find it more convenient to use the fundamental domain

F 0 ={z∈H:0≤R(z)≤ 1 ,|z|≥ 1 |z− 1 |≥ 1 }.

This fundamental domain is shown in Fig. 12.4. It is given by cutting the
fundamental domainF 0 along the lineR(z) = 0 and regluing the left-hand
piece to the right-hand side.
If onFwe now glue the left-hand side to the right-hand side, we obtain a
surfaceS 0 , which is topologically a once-punctured sphere, with two orbifold
points of orders 2 and 3, respectively, corresponding to the pointi∼i+1 and