The kernel of the Goldberg homomorphism is not finitely generated
DOI:
https://doi.org/10.7146/math.scand.a-158031Abstract
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 $\pi _1(M)$ and showed that the kernel of the homomorphism is finitely normally generated. Here we show that the kernel is \emph {not} finitely generated. The proof is an elementary application of covering space theory and the geometry of the euclidean or hyperbolic plane.
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