Compact quotient manifolds of domains in a complex 3-dimensional projective space and the Lebesgue measure of limit sets
type: Item
instance of: scholarly article
title: Compact quotient manifolds of domains in a complex 3-dimensional projective space and the Lebesgue measure of limit sets
zbMATH Open document ID: 0864.57034
author: Masahide Kato
published in: Tokyo Journal of Mathematics
publication date: 1997-06-22T00:00:00Z
review text: The author considers groups acting on the three-dimensional complex projective space \(P^3\) which are the analogs of function groups in the classical theory of Kleinian groups. More precisely, he considers discontinuous groups \(\Gamma< PGL_4 (\mathbb{C})\) acting discontinuously and freely on some domain \(\Omega \subset P^3\) so that \(\Omega/ \Gamma\) is a compact manifold. In the above situation, the domain \(\Omega\) is assumed to contain a subdomain biholomorphically equivalent to an open set of \(P^3\) containing a complex projective line \(P^1\). Two main tools in the theory of Kleinian groups are the Maskit combination theorems, which permit to obtain new Kleinian groups from old ones. One of the main properties of these combination theorems is the fact that the combination of Kleinian groups with Lebesgue measure zero limit sets gives again Kleinian groups with Lebesgue measure zero limit sets. The main result of this paper is the validity of the above property for the groups acting on \(P^3\) as above.
Mathematics Subject Classification ID: 57S30
zbMATH DE Number: 927036
zbMATH Keywords: groups
zbMATH Keywords: three-dimensional complex projective space
zbMATH Keywords: Kleinian groups
reviewed by: Rubén A. Hidalgo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270043222
OpenAlex ID: W2075051561
DOI: 10.3836/TJM/1270043222
description: scientific article; zbMATH DE number 927036