Let
C
C
be a smooth projective curve over a field
k
k
. For each closed point
Q
Q
of
C
C
let
C
=
C
(
C
,
Q
,
k
)
\mathcal {C} = \mathcal {C}(C, Q, k)
be the coordinate ring of the affine curve obtained by removing
Q
Q
from
C
C
. Serre has proved that
G
L
2
(
C
)
GL_2(\mathcal {C})
is isomorphic to the fundamental group,
π
1
(
G
,
T
)
\pi _1(G, T)
, of a graph of groups
(
G
,
T
)
(G, T)
, where
T
T
is a tree with at most one non-terminal vertex. Moreover the subgroups of
G
L
2
(
C
)
GL_2(\mathcal {C})
attached to the terminal vertices of
T
T
are in one-one correspondence with the elements of
Cl
(
C
)
\operatorname {Cl}(\mathcal {C})
, the ideal class group of
C
\mathcal {C}
. This extends an earlier result of Nagao for the simplest case
C
=
k
[
t
]
\mathcal {C} = k[t]
. Serre’s proof is based on applying the theory of groups acting on trees to the quotient graph
X
¯
=
G
L
2
(
C
)
∖
X
\overline {X} = GL_2(\mathcal {C}) \backslash X
, where
X
X
is the associated Bruhat-Tits building. To determine
X
¯
\overline {X}
he makes extensive use of the theory of vector bundles (of rank 2) over
C
C
. In this paper we determine
X
¯
\overline {X}
using a more elementary approach which involves substantially less algebraic geometry. The subgroups attached to the edges of
T
T
are determined (in part) by a set of positive integers
S
\mathcal {S}
, say. In this paper we prove that
S
\mathcal {S}
is bounded, even when Cl
(
C
)
(\mathcal {C})
is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of
G
L
2
(
C
)
GL_2(\mathcal {C})
, involving unipotent and elementary matrices.