  We study a novel type of braid groups on a closed orientable surface
$\Sigma$. These are fundamental groups of certain manifolds that are hybrids
between symmetric products and configuration spaces of points on $\Sigma$; a
class of examples arises naturally in gauge theory, as moduli spaces of
vortices in toric fibre bundles over $\Sigma$. The elements of these braid
groups, which we call divisor braids, have coloured strands that are allowed to
intersect according to rules specified by a graph $\Gamma$. In situations where
there is more than one strand of each colour, we show that the corresponding
braid group admits a metabelian presentation as a central extension of the free
Abelian group $H_1(\Sigma;\mathbb{Z})^{\oplus r}$, where $r$ is the number of
colours, and describe its Abelian commutator. This computation relies crucially
on producing a link invariant (of closed divisor braids) in the three-manifold
$S^1 \times \Sigma $ for each graph $\Gamma$. We also describe the von Neumann
algebras associated to these groups in terms of rings that are familiar from
noncommutative geometry.
