The kernel of the Goldberg homomorphism is not finitely generated

Let M be a closed surface other than the sphere or projective plane. Goldberg defined a natural homomorphism from the n-stranded pure braid group of M to the n-fold product of the fundamental group of M and showed that the kernel of the homomorphism is finitely normally generated. Here we show that the kernel is not finitely generated. The proof is an elementary application of covering space theory and the geometry of the euclidean or hyperbolic plane.