Goldman has constructed a symplectic form on the moduli space
Hom
(
π
,
G
)
/
G
\operatorname {Hom} (\pi ,G)/G
, of flat
G
G
-bundles over a Riemann surface
S
S
whose fundamental group is
π
\pi
. The construction is in terms of the group cohomology of
π
\pi
. The proof that the form is closed, though, uses de Rham cohomology of the surface
S
S
, with local coefficients. This symplectic form is shown here to be the restriction of a tensor, that is defined on the infinite product space
G
π
{G^\pi }
. This point of view leads to a direct proof of the closedness of the form, within the language of group cohomology. The result applies to all finitely generated groups
π
\pi
whose cohomology satisfies certain conditions. Among these are the fundamental groups of compact Kähler manifolds.