  From a group action on a space, define a variant of the configuration space
by insisting that no two points inhabit the same orbit. When the action is
almost free, this "orbit configuration space" is the complement of an
arrangement of subvarieties inside the cartesian product, and we use this
structure to study its topology. We give an abstract combinatorial description
of its poset of layers (connected components of intersections from the
arrangement) which turns out to be of much independent interest as a
generalization of partition and Dowling lattices. The close relationship to
these classical posets is then exploited to give explicit cohomological
calculations.
