On orbit configuration spaces associated to the Gaussian integers: Homotopy and homology groups

The purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from the standard integral lattice Z+iZ in the complex numbers. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to the Lie algebra obtained from the higher homotopy groups of “higher dimensional arrangements” modulo torsion. The resulting Lie algebras are similar to those studied by T. Kohno associated to elliptic KZ systems [Topology Appl. 78 (1997) 79–94]. A question about the generality of this behavior is posed.