  We will define and study some generalisations of pure $\mathfrak{g}$-braid
groups that occur in the theory of connections on curves, for any complex
reductive Lie algebra $\mathfrak{g}$. They make up local pieces of the wild
mapping class groups, which are fundamental groups of (universal) deformations
of wild Riemann surfaces, underlying the braiding of Stokes data and
generalising the usual mapping class groups. We will establish a general
product decomposition for the local wild mapping class groups, and in many
cases define a fission tree controlling this decomposition. Further in type A
we will show one obtains cabled versions of braid groups, related to braid
operads.
