230 R. Brooks
there are other features of the geometry of Riemann surfaces which can be
read off from this picture.
The key point here is to take seriously the additional structure afforded by
the orientation of a graph. As we have seen, the orientation does not have a
strong effect on the behavior ofλ 1 (and indeed has no effect on the behavior
ofλ 1 of a graph), but it does have a very strong effect on the geometry of the
surface.
Indeed, what is the difference between the two pictures of the Ramanujan
graphX^2 ,^3 given in the introduction? The point is that sinceX^2 ,^3 is a homo-
geneous graph, it carries a natural orientation. It was the orientation that was
responsible for unraveling the chaos of the first picture to obtain the order of
the second picture.
In this section, we will describe the following two results:
Theorem 8.Let(Γ,O)be chosen randomly among oriented 3-regular graphs
onnvertices. Then the expected valueE(genus(SC(Γ,O))satisfies
(const) + (n/4)−(3/4) log(n)≤E(genus)≤(const) + (n/4)−(1/2) log(n)Theorem 9.IfSis a Riemann surface, denote byEmb(S)the area of the
largest embedded ball inS. Then the expected value ofEmb(SC(Γ,O))satisfies
E(Emb(SC(Γ,O))≥(1/π)area(S).Of course, the first theorem tells us more about our method of picking
Riemann surfaces then it is about the surfaces themselves, but the second
theorem tells us a fascinating fact about Riemann surfaces – the general Rie-
mann surface has its geometry dominated by one very large embedded ball.
The idea of the proofs of these theorems is to translate them into what
they are stating about LHT paths. According to our formula for the genus,
the first theorem shows that the expected value of the number of LHT paths
grows logarithmically inn. A reasonable question to ask is what one expects
about the associated lengths of the left-hand turn paths. We claim that the
second theorem shows us about the expected length of the longest LHT path:
Lemma 4.Let(Γ,O)be an oriented graph. For given LHT pathC,letLCbe
the length of this path. IfSO(Γ,O)obeys the large cusp condition, then about
the image of the corresponding cusp ofSO(Γ,O)inSC(Γ,O), there exists an
embedded ball of area∼LC.
The proof is to apply the Ahlfors–Schwarz lemma. The horocycle neigh-
borhood ofCinSO(Γ,O) has area equal to the length ofC, and this horocycle
neighborhood goes over to an embedded ball in the compactification.
Before discussing the proofs of these theorems, we will say a few words
concerning where they come from. If we pick an elementσ=σnrandomly
from the symmetric groupS(n)onnelements, we may ask to give thecycle
decompositionofσ. This will of course determineσup to conjugacy. We then
have: