Higher dimensional complex Kleinian groups
type: Item
instance of: scholarly article
title: Higher dimensional complex Kleinian groups
zbMATH Open document ID: 1022.37032
author: Alberto Verjovsky
author: José Seade
published in: Mathematische Annalen
publication date: 2002-05-29T00:00:00Z
full work available at URL: http://cds.cern.ch/record/420234
review text: Let \(N\) be an oriented Riemannian manifold of dimension \(2n\). The twistor space of \(N\) is the total space of a bundle over \(N\), whose fiber at each \(x\in N\) is the symmetric space \(SO(2n)/U(n)\), of all complex structures on \(T_x(N)\) which are compatible with the metric and the orientation of \(N\). The work of R. Penrose in the 1970's established a beautiful interplay between the conformal geometry of Riemannian manifolds and the complex geometry of their twistor spaces. In the article under review, the authors push forward these ideas and use twistor theory to construct new complex Kleinian groups in higher dimensions. One of their themes is that the conformal dynamics of Kleinian groups on spheres embeds in the holomorphic dynamics of complex Kleinian groups on projective spaces. They notice that if \(\Gamma\) is a discrete group of orientation-preserving conformal diffeomorphisms of \(N\), then \(\Gamma\) has a canonical lifting to a group \({\widetilde \Gamma}\) of diffeomorphisms of the twistor space of \(N\). They show that there is a natural limit set for the action of \({\widetilde \Gamma}\), which is the inverse image of the limit set of \(\Gamma\). They note an interesting feature here, which does not happen in the classical theory of limit sets of conformal Kleinian groups, which is the fact that the action of \({\widetilde \Gamma}\) on its limit set may or may not be minimal and ergodic with respect to the densities which come from the Patterson-Sullivan densities for \(\Gamma\). In the meantime, the authors prove a generalized version of a theorem which originates in the work of Sullivan on conformal dynamics and which says that in the case of the classical Kleinian groups, there is no invariant line field on the limit set. The authors' generalization says that every geometrically finite Zariski-dense subgroup of conformal maps of the sphere \(S^n\) acts minimally and ergodically on the \(p\)-dimensional subspaces of \(T(S^n)\) over the li…
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 37F30
Mathematics Subject Classification ID: 37F40
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1747290
zbMATH Keywords: Kleinian group
zbMATH Keywords: conformal dynamics
zbMATH Keywords: complex Kleinian group
zbMATH Keywords: twistor
zbMATH Keywords: Patterson-Sullivan measure
zbMATH Keywords: limit set
MaRDI profile type: Publication
OpenAlex ID: W2092294618
DOI: 10.1007/S002080100247
description: scientific article; zbMATH DE number 1747290