12 The Spectral Geometry of Riemann Surfaces 219Fig. 12.6.The graph on two verticesSO(Γ,O) as a covering space ofH^2 /P SL(2,Z), with the vertex being an orbi-
fold point of order 3.
We remark that the surface so constructed depends in a very heavy way
on the orientation. The easiest way to see this is by seeing on the genus.
According to Gauss–Bonnet, the genus ofSC(Γ,O) can be computed by
genus (SC(Γ,O)) = 1 +n− 2 LHT
4.
This gives us as an amusing sidelight that the number of LHT paths must
have the same parity. Let us take some simple examples of this construction.
We first take the simplest graph, the three-regular graph on two vertices
with no loops or double edges. It has two possible orientations.
Let us now build the surfaceSC(Γ,O). We begin by gluing two triangles
together, as shown below:
In order to glue the remaining two pairs of sides, we need to use the
orientation. With the first orientation, the left-hand side is glued to the top,
while the right-hand side is glued to the bottom. The resulting surface is a
sphere with three punctures, so its compactification must be the sphere.
With the second orientation, the left-hand side is glued to the right-hand
side, while the top is glued to the bottom. We obtain in this way a torus with
one cusp.
Which torus is it? The compactification process tells us that there is a
unique way of assigning angles to the corners, so that first of all the angle
sum around a cusp is 2π, and second so that the tick marks continue to be
glued to tick marks. It is easy to see that assigning an angle ofπ/3toeach
corner fulfills this requirement, because in an equilateral triangle the conformal
midpoint is also the geometric midpoint.
Thus the surfaceSC(Γ,O)equilateral torus, obtained by gluing opposite
sides of a parallelogram, is obtained by gluing two equilateral tori together.
Now let us consider the complete graph on four vertices, also known as
the tetrahedron (Fig. 12.7). It has three essentially distinct orientations — you