We introduce the notion of volume of the representation variety of a finitely
presented discrete group in a compact Lie group using the push-forward measure
associated to a map defined by a presentation of the discrete group. We show
that the volume thus defined is invariant under the Andrews-Curtis moves of the
generators and relators of the discrete group, and moreover, that it is
actually independent of the choice of presentation if the difference of the
number of generators and the number of relators remains the same. We then
calculate the volume of the representation variety of a surface group in an
arbitrary compact Lie group using the classical technique of Frobenius and
Schur on finite groups. Our formulas recover the results of Witten and Liu on
the symplectic volume and the Reidemeister torsion of the moduli space of flat
G-connections on a surface up to a constant factor when the Lie group G is
semisimple.Comment: 27 pages in AMS-LaTeX forma
