Hyperbolic elements in negatively curved groups
type: Item
instance of: scholarly article
title: Hyperbolic elements in negatively curved groups
zbMATH Open document ID: 0834.20041
DOI: 10.1007/BF01264930
author: Eric L. Swenson
published in: Geometriae Dedicata
publication date: 1996-03-18T00:00:00Z
review text: The paper shows that, in a negatively curved group \(G\) (= Gromov hyperbolic group, see \textit{M. Gromov} [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]), the conjugacy class of any infinite cyclic subgroup contains a straight element, that is an element \(g\in G\) with \(|g^n|=n|g|\). To prove this fact, the author uses \textit{J. Cannon's} approach [in Hyperbolic geometry and ergodic theory 1989, 315-369 (1991; Zbl 0764.57002)] to negatively curved groups and Cannon's results on cone types and quasigeodesics in such groups. As a corollary, it is obtained that the translation number of an element of a negatively curved group is rational with uniformly bounded (in the group) denominator. This implies that the endpoints of conjugates of a hyperbolic element are dense in the boundary of the group. This resembles the situation with the limit set of Kleinian groups and allows the author to get an extension of another result from Kleinian groups, namely that every normal subgroup of a negatively curved group is either elementary (finite) or has the same boundary at infinity as the group \(G\) itself. The author also proves the double density theorem of Gromov (see the above reference) that any two boundary points in \(\partial G\) are ``almost'' the endpoints of a hyperbolic element.
reviewed by: Boris N. Apanasov
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 806229
zbMATH Keywords: infinite cyclic subgroups
zbMATH Keywords: straight elements
zbMATH Keywords: negatively curved groups
zbMATH Keywords: cone types
zbMATH Keywords: quasigeodesics
zbMATH Keywords: translation numbers
zbMATH Keywords: Kleinian groups
zbMATH Keywords: boundary at infinity
zbMATH Keywords: double density theorem
zbMATH Keywords: hyperbolic elements
MaRDI profile type: Publication
cites work: Q4865833
cites work: A combination theorem for negatively curved groups
cites work: The Boundary of Negatively Curved Groups
cites work: The combinatorial structure of cocompact discrete hyperbolic groups
cites work: Q4010477
cites work: Q3772406
cites work: Q4039828
full work available at URL: https://doi.org/10.1007/bf01264930
OpenAlex ID: W2044092288
description: scientific article; zbMATH DE number 806229