The kernel of the Goldberg homomorphism is not finitely generated

TL;DR

This paper proves that the kernel of Goldberg's homomorphism from the pure braid group of a closed surface (not sphere or projective plane) to a product of fundamental groups is not finitely generated, contrasting previous results.

Contribution

It demonstrates that the kernel of Goldberg's homomorphism is not finitely generated, providing a new insight into the structure of braid groups on surfaces.

Findings

Kernel is not finitely generated

Uses covering space theory and geometry of Euclidean/hyperbolic plane

Contrasts with previous finite generation results

Abstract

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.