The kernel of the Goldberg homomorphism is not finitely generated - Scharlemann, Martin (MATHEMATICA SCANDINAVICA, 2025)
Let M be a closed surface not the sphere or projective plane. Goldberg [Math. Scand. 33 (1973) 69–82] defined a natural homomorphism from the n-stranded pure braid group of M to the n-fold product of π _1(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.