Volume of representation varieties

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

Algebraic geometric invariants for a class of one-relator groups

AbstractGiven a finitely generated group H, the set Hom(H, SL2C) inherits the structure of an affine algebraic variety R(H) called the representation variety of H. Let a one-relator group with presentation G = 〈x1, …, xn, y; W(x̄) = yk〉 be given, where W(x̄) ≠ 1 is in the free group on the generators {x̄} = {x1, …, xn}, and k ≥ 2. In this paper a theorem will be proven allowing the computation of Dim(R(G)) in terms of subvarieties of the representation variety of the free group on n generators, R(Fn), arising from solutions to the equation W(x̄) = ± l in SL2C. Conditions are given guaranteeing the reducibility of R(G). Finally, applications to the class of one-relator groups with non-trivial center are made