| Description |
A (complex) projective structure is an atlas of charts on a surface S to the Riemann sphere ? whose transition maps are M¨obius transformations. We may think of S as a Riemann surface with the added notion of round circles. A projective structure on S determines a conformal immersion (developing map) of the universal cover, D : ? ? ?, and a (holonomy) representation of the fundamental group ? : rr1(S) ? PSL2(?), both defined up to composition with PSL2(?). The pair (D, ?) globalizes the local information given by the charts and transition functions. We wish to find to what degree global information (a fixed holonomy representation) determines local information (the projective structure). Projective structures are of interest themselves and also arise in hyperbolic geometry since the conformal boundary of a geometrically finite hyperbolic 3-manifold has a projective structure. It is known that (almost) all representations of the fundamental group of a surface in PSL2(?) arise as the holonomy representation of a projective structure on S. It is unknown, however, to what degree the representation characterizes the projective structure. Goldman used a general procedure called grafting to classify all projective structures with faithful holonomy representation onto a quasifuchsian group. Goldman proved that all projective structures with faithful Fuchsian holonomy were uniquely obtained by grafting a standard structure. We first consider projective structures with holonomy a discrete subgroup of PSL2(R). We show such projective structures are distinguished by the homotopy type of certain so-called real curves on the surface. Using this, we formulate a new proof of the aforementioned theorem of Goldman. We then consider the case when ! is a Schottky subgroup of PSL2(R). We provide a characterization for grafting along a certain class of curves. It follows that in suitable situations the effect of grafting on the real curves is a Dehn twist of the initial real curves. We also show that there are infinitely many different ways to graft two standard structures that differ by an elementary move. |
