In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation.
By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface
R
which is not isomorphic to either the Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of the hyperbolic plane
H
by a subgroup
Γ
acting properly discontinuously and freely.
In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformation is the group
P
S
L
2
(
R
)
acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup
Γ
⊂
P
S
L
2
(
R
)
such that the Riemann surface
Γ
∖
H
is isomorphic to
R
. Such a group is called a Fuchsian group, and the isomorphism
R
≅
Γ
∖
H
is called a Fuchsian model for
H
.
Fuchsian models and Teichmüller space
Let
R
be a closed hyperbolic surface and let
Γ
be a Fuchsian group so that
Γ
∖
H
is a Fuchsian model for
R
. Let
A
(
Γ
)
=
{
ρ
:
Γ
→
P
S
L
2
(
R
)
:
ρ
is faithful and discrete
}
and endow this set with the topology of pointwise convergence (sometimes called "algebraic convergence"). In this particular case this topology can most easily be defined as follows: the group
Γ
is finitely generated since it is isomorphic to the fundamental group of
R
. Let
g
1
,
…
,
g
r
be a generating set: then any
ρ
∈
A
(
Γ
)
is determined by the elements
ρ
(
g
1
)
,
…
,
ρ
(
g
r
)
and so we can identify
A
(
G
)
with a subset of
P
S
L
2
(
R
)
r
by the map
ρ
↦
(
ρ
(
g
1
)
,
…
,
ρ
(
g
r
)
)
. Then we give it the subspace topology.
The Nielsen isomorphism theorem (this is not standard terminology and this result is not directly related to the Dehn-Nielsen theorem) then has the following statement:
For any
ρ
∈
A
(
G
)
there exists a self-homeomorphism (in fact a quasiconformal map)
h
of the upper half-plane
H
such that
h
∘
γ
∘
h
−
1
=
ρ
(
γ
)
for all
γ
∈
G
.
The proof is very simple: choose an homeomorphism
R
→
ρ
(
Γ
)
∖
H
and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since
R
is compact.
This result can be seen as the equivalence between two models for Teichmüller space of
R
: the set of discrete faithful representations of the fundamental group
π
1
(
R
)
into
P
S
L
2
(
R
)
modulo conjugacy and the set of marked Riemann surfaces
(
X
,
f
)
where
f
:
R
→
X
is a quasiconformal homeomorphism modulo a natural equivalence relation.
