Hyperbolic structures on the configuration space of six points in the projective line

The oriented configuration space X-6(+) of six points on the real projectiv e line is a noncompact three-dimensional manifold which admits: a unique co mplete hyperbolic structure of finite volume with ten cusps. On the other h and, it decomposes naturally into 120 cells each of which can be interprete d as the set of equiangular hexagons with unit area. Similar hyperbolic str uctures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X-6(+) is at the center of a live paramete r family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhib it two real analytic maps from the set of prescribed angles of hexagons int o R-10 whose components are the traces of the monodromics at the ten cusps. We show that this map has maximal rank 5 at the center. (C) 2000 Academic Press.