10Mobius group

IA Groups 10.2 Fixed points of obius maps
Definition (Fixed point). A fixed point of f is a z such that f(z) = z.
We know that any obius map with
c
= 0 fixes
. We also know that
z z
+
b
for any
b 6
= 0 fixes
only, where as
z 7→ az
for
a 6
= 0
,
1 fixes 0 and
. It turns out that you cannot have more than two fixed points, unless you
are the identity.
Proposition.
Any obius map with at least 3 fixed points must be the identity.
Proof.
Consider
f
(
z
) =
az+b
cz+d
. This has fixed points at those
z
which satisfy
az+b
cz+d
=
z cz
2
+ (
d a
)
z b
= 0. A quadratic has at most two roots, unless
c = b = 0 and d = a, in which the equation just says 0 = 0.
However, if c = b = 0 and d = a, then f is just the identity.
Proposition.
Any obius map is conjugate to
f
(
z
) =
νz
for some
ν 6
= 0 or to
f(z) = z + 1.
Proof.
We have the surjective group homomorphism
θ
:
GL
2
(
C
)
M
. The
conjugacy classes of GL
2
(C) are of types
λ 0
0 µ
7→ g(z) =
λz + 0
0z + µ
=
λ
µ
z
λ 0
0 λ
7→ g(z) =
λz + 0
0z + λ
= 1z
λ 1
0 λ
7→ g(z) =
λz + 1
λ
= z +
1
λ
But the last one is not in the form
z
+ 1. We know that the last
g
(
z
) can
also be represented by
1
1
λ
0 1
, which is conjugate to
1 1
0 1
(since that’s its
Jordan-normal form). So z +
1
λ
is also conjugate to z + 1.
Now we see easily that (for
ν 6
= 0
,
1),
νz
has 0 and
as fixed points,
z
+ 1
only has . Does this transfer to their conjugates?
Proposition. Every non-identity has exactly 1 or 2 fixed points.
Proof.
Given
f M
and
f 6
=
id
. So
h M
such that
hfh
1
(
z
) =
νz
. Now
f
(
w
) =
w hf
(
w
) =
h
(
w
)
hfh
1
(
h
(
w
)) =
h
(
w
). So
h
(
w
) is a fixed point
of
hfh
1
. Since
h
is a bijection,
f
and
hfh
1
have the same number of fixed
points.
So
f
has exactly 2 fixed points if
f
is conjugate to
νz
, and exactly 1 fixed
point if f is conjugate to z + 1.
Intuitively, we can show that conjugation preserves fixed points because if we
conjugate by
h
, we first move the Riemann sphere around by
h
, apply
f
(that
fixes the fixed points) then restore the Riemann sphere to its original orientation.
So we have simply moved the fixed point around by h.