Orbit configuration spaces associated to discrete subgroups of PSL

The purpose of this article is to analyze several Lie algebras associated to " orbit configuration spaces " obtained from a group G acting freely, and properly discontinuously on the upper 1/2-plane À 2. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to 1. the Lie algebra obtained from the higher homotopy groups of " higher dimensional arrangements " modulo torsion, as well as 2. the Lie obtained from horizontal chord diagrams for surfaces. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by G is also addressed.