Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

Chapter 2
Hyperbolic Geometry
2.1 Möbius transformations
The reflection (or inversion) in the sphere S(a, r) = {x ∈ R n : |x − a| = r} where a ∈ R n and
r > 0, is a function φ(·) such that, for a, x ∈ R n , x = a
φ(x) = a +
r2
(x − a), (2.1)
|x − a| 2
it is natural to define φ(a) = ∞ and φ(∞) = a. We observe that φ(x) = x if and only if x ∈ S(a, r).
We have that φ ◦ φ(x) = x and so φ(·) is a bijection of R n onto itself.
A plane P (a, t) in R n is a set of the form
P (a, t) = {x ∈ R n : 〈x, a〉 = t} ∪ {∞}
where a ∈ R n , a = 0, t ∈ R and 〈x, a〉 = aixi. The reflection φ(·) in the plane P (a, t) is defined,
for every x ∈ R n , by
φ(x) = x + λa
and φ(∞) = ∞. The parameter λ is chosen so that 1
2 (x + φ(x)) ∈ P (a, t), this gives the explicit
formula
φ(x) = x − 2a
(〈x, a〉 − t) .
|a| 2 (2.2)
In fact 1
λa
λ
2 (x + φ(x)) ∈ P (a, t) if and only if 〈x + 2 , a〉 = t that is 〈x, a〉 + 2 |a|2 =t. Again φ(x) = x
if and only if x ∈ P (a, t), we have φ ◦ φ(x) = x and so φ(·) is a bijection of R n onto itself.
It is possible to prove that the reflection in a sphere and the reflection in a plane are two continuous
functions in R n onto itself with respect to the chordal metric d(x, y) defined by
⎧
⎨
d(x, y) =
⎩
2|x−y|
(1+|x| 2 ) 1/2 (1+|y| 2 ) 1/2 , if x, y = ∞,
2
(1+|x| 2 ) 1/2 , if y = ∞.