218 R. Brooks
12.5 The Basic Construction
Let Γ be a three-regular graph. An orientationOon Γ is an assignment, for
each vertexvof Γ, of a cyclic ordering of the edges coming out of each vertex.
There is no “compatibility” requirement, so that a graph onnvertices will
have in general 2norientation.
To the pair (Γ,O), will assign a Riemann surface. Actually, we will assign
two Riemann surfaces,SO(Γ,O)andSC(Γ,O). The surfaceSO(Γ,O) will be a
finite area Riemann surface, while the surfaceSC(Γ,O) will be the conformal
compactification ofSO(Γ,O) obtained by filling in each cusp with a point.
We begin by considering the ideal hyperbolic triangleT(Fig. 12.5), which
is the unique triangle with verticles at 0, 1 ,∞. We will mark some points on
T– we will marks at the pointsi,i+ 1, and (i+1)/2, which can be thought
of as the midpoints of the sides. We then join these points by horocycles. We
then consider the point (1 +i
√
3)/2, and draw the geodesics joining this point
to the three midpoints. To finish things up, we will draw a “traffic pattern”
onT, showing the cyclic ordering corresponding to always turning left.
Here is what it looks like:
To the pair (Γ,O) we construct the surfaceSO(Γ,O) in the following way:
at each vertex, we place a copy ofT, so that the traffic pattern onTmatches
up with the orientation at the vertex. Whenever two vertices are joined by
an edge, we glue the corresponding triangles together so that the tick marks
are glued together and the orientations match up (Fig. 12.6). This describes
a unique gluing procedure.
We remark that the horocycle pieces on eachTglue together to give closed
horocycles about a cusp. Indeed, each cusp on the surfaceSO(Γ,O) corre-
sponds to a path on the graph such that each time you arrive at a vertex, you
turn left. We will call such pathsleft-hand-turnpaths, or LHT paths for short.
It is easy to see that each surfaceSC(Γ,O) is a Belyi surface, and con-
versely, each Belyi surface arises this way. The oriented graph (Γ,O) describes
Fig. 12.5.The marked-up ideal triangleT