A characterization of extended Schottky type groups with a remark to Ahlfors' conjecture
type: Item
instance of: scholarly article
title: A characterization of extended Schottky type groups with a remark to Ahlfors' conjecture
zbMATH Open document ID: 0727.30035
author: Katsuhiko Matsuzaki
published in: Journal of Mathematics of Kyoto University
publication date: 1991-01-01T00:00:00Z
review text: The author considers the class of extended Schottky groups. Informally, these are like Schottky groups where some of the loxodromics are allowed to be parabolic and there may be some rank two parabolic subgroups. He shows that a finitely generated, torsion-free Kleinian group is extended Schottky precisely when its limit set is totally disconnected. Equivalently, it is a function group which is constructible from elementary groups using combination theorems. (Here one must be careful to avoid any Fuchsian subgroups, but the techniques are well-known constructions due to Maskit.) He also shows that a Kleinian group is an extended Schottky group if it is a free product of a finite number of infinite cyclic groups and a finite number of rank two parabolics. As an application of these ideas and the well-known Bonahon condition, he shows that a 2-generator torsion-free Kleinian group has a limit set of zero area.
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 4200570
zbMATH Keywords: hyperbolic 3-manifolds
zbMATH Keywords: area-type inequalities
zbMATH Keywords: extended Schottky groups
zbMATH Keywords: torsion-free Kleinian group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519903
OpenAlex ID: W1480133169
Wikidata QID: Q123333779
DOI: 10.1215/KJM/1250519903
description: scientific article; zbMATH DE number 4200570