Representations of surface groups in complex hyperbolic space
type: Item
instance of: scholarly article
title: Representations of surface groups in complex hyperbolic space
zbMATH Open document ID: 0676.57012
author: Domingo Toledo
published in: Journal of Differential Geometry
publication date: 1989-01-01T00:00:00Z
review text: Let S be a closed Riemann surface of genus \(g>1\). If \(\rho\) : \(\pi_ 1S\to PSU(1,n)\) is a representation of the surface group, the associated flat bundle over S fibered by \(B^ n\) can be trivialised by means of an equivariant map f: \(H^ 2\to B^ n\). If \(\omega\) is the natural Kähler form on \(B^ n\), let \(f^*\omega\) denote the induced form on S, and write \(c(\rho)=\int_{S}f^*\omega\). Then it is known that \(| c(\rho)| \leq 4\pi (g-1),\) and the author proves that if we have equality, then \(\rho\) leaves invariant some complex line in \(B^ n\). He deduces that the space of flat bundles such that \(| c(\rho)| -4\pi (g-1)\) is homeomorphic to the product of the Teichmüller space for S and the space of flat U(n-1)-bundles over S. The critical step in the proof is the extension of f to a map of \(\bar H^ 2\) into \(\bar B^ n\); the image of \(\partial \bar H^ 2\) in \(\bar B^ n\) is then the boundary of the required complex line.
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 20F99
zbMATH DE Number: 4107754
zbMATH Keywords: representation of fundamental group by hyperbolic isometries
zbMATH Keywords: Riemann surface
zbMATH Keywords: flat bundle
zbMATH Keywords: Kähler form
zbMATH Keywords: Teichmüller space
zbMATH Keywords: flat U(n-1)- bundles
reviewed by: Charles B. Thomas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442638
OpenAlex ID: W1488772207
Wikidata QID: Q115182921
DOI: 10.4310/JDG/1214442638
description: scientific article; zbMATH DE number 4107754