  We generalize presentations of the fundamental group of discriminant
complements and arrive at a class of presentations associated naturally with
words in the free monoid of the alphabet $\sigma_1,\dots,\sigma_{n-1}$. Our
study addresses invariance properties of these presentations and the presented
groups under various operations on the words. In particular we prove that the
group does only depend on the corresponding element in the positive braid
monoid, and under mild hypotheses only on its conjugacy class in the braid
group.
