Orbit configuration spaces associated to discrete subgroups of PSL(2,R)PSL(2,R)

The purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from a group GG acting freely, and properly discontinuously on the upper half-plane H2H2. 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 analogous constructions associated to H2×CqH2×Cq modulo torsion, as well as2.the Lie algebra obtained from horizontal chord diagrams for surfaces. The resulting Lie algebras are similar to those studied in [T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math. 78 (1988) 339–363; T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math. 179 (1994) 123–138; T. Kohno, Elliptic KZ system, braid groups of the torus and Vassiliev invariants, Topology and its Applications 78 (1997) 79–94; D.C. Cohen, Monodromy of fiber-type arrangements and orbit configuration spaces, Forum Math. 13 (2001) 505–530; F.R. Cohen, M. Xicoténcatl, On orbit configuration spaces associated to the Gaussian integers: homology and homotopy groups, Topology Appl. 118 (2002) 17–29; E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. J. Julius Schauder Center 11 (1998), 249–271; E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, in: Springer Monographs in Mathematics, Springer-Verlag, 2001; F.R. Cohen and T. Sato, On groups of morphisms of coalgebras, (submitted for publication)]. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by GG is also addressed.