1. Mobius transformations
The motion A has either one or two fixed points a, (3 in C; they are given
by the formulae
{a, / 3 } = | l ( ( a - d ) ± ^ t r 2 ( ^ ) - 4 ) | if c ± 0,
an d
{a, /?} = ^ oo5 if c = 0.
[ a
a~l
J
It follows from the above formulae7 that A is parabolic if and only if
A has one fixed point in C, if and only if A is conjugate in PSL(2,C) to
. In all other cases A ^ I has two fixed points and is hence conju-
gate in PSL(2,C) to
elliptic if and only if
K
0
0
K-1
with « G C - { 0 , 1 , - 1 } . In this case A is
K\ = 1, A is hyperbolic if and only if K R, and A is
loxodromic if and only if \K\ 1.
The square of «, n2{A) is called the multiplier of A The multiplier
is defined uniquely except that a number and its reciprocal are the two
multipliers of the same Mobius
transformation.8
For A parabolic or the
identity, we set K2(A) = 1. We note that
K2(An)
=
(n2(A))n,
for all A e PSL(2,C).
Hence only elliptic elements can have finite order. A Mobius transformation
is of finite order if and only if its multiplier is a root of unity. The order n
of such a motion A is the smallest positive integer n such that
K2n{A)
= 1.
The trace and the square root of the multiplier are related by the formula
tr =
K,
+
K
- 1
.
A parabolic motion A with finite fixed point a and translation length a can
be written in normal form as a motion of the sphere as
1 1
—— = f - a
A\z) a z a
(with a E C*), or equivalently as an element of PSL(2,C) as
1 + aa
A
2
-aza
a aa
7Many times hereafter we will derive onlyformulae for finite values of the parameters (or vari-
ables) (here fixed points). We will usually leave it to the reader to make the necessary adjustments
for the parameter assuming the value oo.
8For loxodromic motions, it is always possible to choose the multiplier so that its magnitude
is 1. For primitive elliptic motions we may choose the multiplier so that its argument lies in
(0,4
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