The Maximal Number of Transverse Self-Intersections of Geodesics ...

ackground is as stated in [CDGISW]. First, we will review some hyperbolic geometry.
The hyperbolic upper half plane, H, isdefinedontheset{x + iy : y>0}. Geodesics on H
are either semicircles centered on the horizontal axis or infinite vertical lines. We can use
the group ½µ
¾
a b
SL(2, Z) =
: a, b, c, d ∈ Z,ad− bc =1 to act on H through the homomorphism
defined by
c d
µ a b
T = 7→ Tz = az + b
c d
cz + d .
This group of fractional linear transformation is Γ = PSL(2, Z). Let Γ 0 be the commutator
subgroup of Γ. Γ 0 isasubgroupontwogenerators
µ µ
1 1
1 −1
a = and b =
. The following definition is from [CR].
1 2
−1 2
Definition 1.1 ThefreegroupF 2 on two generators is the group with generators a, b in which
no relation exists except the trivial one between an element x and its inverse X.
Notation 1.2 Let T be a once-punctured torus.
We can now look at a fundamental region defined on H. We let the fundamental region
be the points in H above the semicircle with feet at 0 and 1 and the semicircle with feet at
−1 and 0, and between the infinite vertical lines −1 and 1. The following diagram shows
the fundamental region with a and b drawn in.
1
0.8
0.6
0.4
0.2
–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1
x
2