Projective structures with Fuchsian holonomy.

*(English)*Zbl 0595.57012This paper discusses the classification of real and complex projective structures on closed surfaces whose holonomy homomorphism is a fixed Fuchsian representation (i.e. a discrete faithful representation of the fundamental group into PSL(2, \({\mathbb{C}})\). The basic construction, due independently to Maskit, Hejhal, Sullivan-Thurston, inserts annuli into hyperbolic structures creating complex projective structures whose holonomy homomorphisms remain Fuchsian yet the developing maps become surjective maps onto the complex projective line. In particular the developing maps fail to be covering maps onto their image. It is shown that every projective structure whose holonomy is a Fuchsian representation arises from this construction. A similar theorem is given for real projective structures, where the construction is more complicated. It is also shown how to construct real and complex projective structures with various non-Fuchsian holonomy representations. Finally an analogous construction for flat conformal structures in higher dimensions is given, showing the existence of flat conformal structures on closed hyperbolic manifolds with surjective developing maps.

##### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

55R99 | Fiber spaces and bundles in algebraic topology |

53C05 | Connections, general theory |

57N15 | Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) |

57R30 | Foliations in differential topology; geometric theory |