On discrete isometry groups of negative curvature
type: Item
instance of: scholarly article
title: On discrete isometry groups of negative curvature
zbMATH Open document ID: 0822.57026
author: Gaven J. Martin
published in: Pacific Journal of Mathematics
publication date: 1995-10-17T00:00:00Z
review text: In the paper under review discrete isometry groups of a Hadamard manifold all of whose sectional curvatures are pinched by \(-1\) and \(-a^ 2\) are called negatively curved groups. The author extends several well-known results in the theory of Kleinian groups (the case of constant negative curvature) like Jørgensen inequality and Jørgensen's algebraic limit theorem to the case of negatively curved groups. One of the results is: let \(\Gamma_ n\) be a sequence of finitely generated isomorphic negatively curved nonelementary groups, acting on Hadamard spaces with uniformly pinched sectional curvatures, converging algebraically to a group \(\Gamma_ \infty\) of homeomorphisms of \(\mathbb{B}^ n\). If in addition the sequence \(\Gamma_ n\) has uniformly bounded torsion (all elements of finite order of all groups \(\Gamma_ n\) have orders bounded by the same finite constant) then the limit group \(\Gamma_ \infty\) is discrete and is isomorphic to \(\Gamma_ n\) (\(n \in \mathbb{N}\)). The condition on the torsion subgroup is essentially even in the Kleinian group case according to examples due to the author [Acta Math. 163, No. 3/4, 253-289 (1989; Zbl 0698.20037)].
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 572467
zbMATH Keywords: discrete isometry groups of a Hadamard manifold
zbMATH Keywords: negatively curved groups
zbMATH Keywords: Jørgensen inequality
zbMATH Keywords: Jørgensen's algebraic limit theorem
reviewed by: Leonid Potyagailo
MaRDI profile type: Publication
full work available at URL: pjm.1993.160.109
OpenAlex ID: W2012130437
DOI: 10.2140/PJM.1993.160.109
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description: scientific article; zbMATH DE number 572467