Minimal index torsion-free subgroups of Kleinian groups
type: Item
instance of: scholarly article
title: Minimal index torsion-free subgroups of Kleinian groups
zbMATH Open document ID: 0890.57016
author: K. N. Jones
author: Alan W. Reid
published in: Mathematische Annalen
publication date: 1998-05-25T00:00:00Z
review text: It was shown by \textit{A. L. Edmonds, J. H. Ewing} and \textit{R. S. Kulkarni} [Invent. Math. 69, 331-346 (1982; Zbl 0498.20033)] that the minimal index of a torsion-free subgroup of a finitely generated Fuchsian group of the first kind \(G\) is bounded above by twice the LCM of the orders of the finite subgroups of \(G\). Here the authors show that no such result is possible for Kleinian groups. Specifically, they exhibit a sequence \(\Gamma_k\) of co-compact Kleinian groups for which the ratio of the minimum index to the LCM is arbitrarily large. The construction of the \(\Gamma_k\) uses generalized triangle groups and unknotting tunnels of 2-bridge knots. The authors also derive some results of independent interest involving these two constructs.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 1143000
zbMATH Keywords: LCM
zbMATH Keywords: generalized triangle groups
zbMATH Keywords: unknotting tunnels of 2-bridge knots
MaRDI profile type: Publication
full work available at URL: s002080050147
OpenAlex ID: W2037363056
DOI: 10.1007/S002080050147
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description: scientific article; zbMATH DE number 1143000