Projective structures with Fuchsian holonomy
type: Item
instance of: scholarly article
title: Projective structures with Fuchsian holonomy
zbMATH Open document ID: 0595.57012
author: William M. Goldman
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: This 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.
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 55R99
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 3958243
zbMATH Keywords: real and complex projective structures on closed surfaces
zbMATH Keywords: holonomy homomorphism
zbMATH Keywords: Fuchsian representation
zbMATH Keywords: developing maps
zbMATH Keywords: flat conformal structures
zbMATH Keywords: foliated bundle
zbMATH Keywords: flat connection
zbMATH Keywords: Teichmüller space
zbMATH Keywords: measured lamination space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214440978
OpenAlex ID: W1520325860
Wikidata QID: Q115184813
DOI: 10.4310/JDG/1214440978
description: scientific article; zbMATH DE number 3958243