The Arithmetic of Quaternion Algebra

86 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS
It follows two consequences:
1) If c �= 0, the location of points such that |dt| = |dz| is the circle |cz + d| =
1. the circle is called the isometric circle of the homography, which play an
important role in the construction of fundamental domain explicitly of discrete
subgroup Γ ⊂ P SL(2, R) in H.
2) P SL(2, R) acts on H by the isometry of the upper half-plane H equipped
with its hyperbolic metrics. The isotropic group of point i = (01) in SL(2, R)
is SO(2, R). The action of P SL(2, R) on H is transitive. We have then a
realization:
H = SL(2, R)/SO(2, R).
W e thus can talk about length, area, geodesic for the hyperbolic metrics on H.
WE obtain
Definition 4.5. The hyperbolic lengthof a curve in H is the integral
�
|dz|y −1 ,
taking along this curve.
The hyperbolic surface of an area of in H is the double integral
� �
y −2 dxdy,
taking in the interior of this area.
The hyperbolic geodesic are the circles with its center at the real axis (including
the line perpendicular to the real axis).
(Here is a picture)!!!
The real axis is the line to the infinity of H.
The isometric group of H is isomorphic to P GL(2, R). We associate to
GL(2, R) the homography
�
(az + b)(cz + d)
t =
−1 , if ad − bc > 0
(a¯z + b)(c¯z = d) −1 , if ad − bc < 0.
� �
a b
∈
c d
Proposition 4.2.1. 2.1. The hyperbolic distance of two points z1, z2 ∈ H is
equal to
d(z1, z2) = |arccosh(1 + |z1 = z2| 2 /2z1z2|.
Proof. (Here is a picture!!!)
If the geodesic between the two points is a vertical line,ds = | � y2
dy/y| =
y1
| log(y2/y1)|. If the geodesic is an arc of the circle with center on the real axis,
� � θ2
ds = θ1 dθ/ sin θ = | log |tg(θ1/2)/tg(θ2/2)||. In all of these two cases we find
that given formula.