In this paper, we study the pro-$\Sigma$ fundamental groups of configuration, where $\Sigma$ is either the set of all prime numbers or a set consisting of a single prime number. In particular, we show, via two somewhat distinct approaches, that, in many cases, the "fiber subgroups" of such fundamental groups arising from the various natural projections of a configuration space to lower-dimensional configuration spaces may be characterized group-theoretically.
"The algebraic and anabelian geometry of configuration spaces." Hokkaido Math. J. 37 (1) 75 - 131, February 2008. https://doi.org/10.14492/hokmj/1253539588