The equivalence of some residual properties of word-hyperbolic groups
type: Item
instance of: scholarly article
title: The equivalence of some residual properties of word-hyperbolic groups
zbMATH Open document ID: 0951.20029
author: Ilya Kapovich
author: Daniel T. Wise
published in: Journal of Algebra
publication date: 2000-09-25T00:00:00Z
review text: We recall that a group \(G\) is said to be residually finite if for each non-trivial element \(g\in G\), there is a finite quotient \(G\to\overline G\) such that the image of \(g\) in \(\overline G\) is non-trivial. Important open questions on word-hyperbolic groups are whether each such group is residually finite, and whether each such group is linear. (By a result of Mal'cev, an affirmative answer to the second question implies an affirmative answer to the first one.) The main result of this paper is the following Theorem. The following are equivalent: (a) Every word-hyperbolic group is residually finite. (b) Every non-trivial word-hyperbolic group has a non-trivial finite quotient.
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1423886
zbMATH Keywords: word-hyperbolic groups
zbMATH Keywords: residually finite groups
zbMATH Keywords: finite quotient groups
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: jabr.1999.8104
OpenAlex ID: W2057847072
cites work: Q4865833
cites work: A combination theorem for negatively curved groups
cites work: Automatic groups and amalgams
cites work: A small simplification in hyperbolic groups
cites work: Statistical properties of finitely presented groups
cites work: Distinguished subgroups and quotients of hyperbolic groups
cites work: ON THE COMBINATION THEOREM FOR NEGATIVELY CURVED GROUPS
cites work: Q4012143
cites work: Q3772406
cites work: Sur les groupes hyperboliques d'après Mikhael Gromov. (On the hyperbolic groups à la M. Gromov)
cites work: Hyperbolic groups and their quotients of bounded exponents
cites work: Hyperbolic groups and free constructions
cites work: Q4867098
cites work: Q4145882
cites work: Q5513066
cites work: Q4273705
cites work: Q3994685
cites work: ALMOST EVERY GROUP IS HYPERBOLIC
cites work: ON RESIDUALING HOMOMORPHISMS AND G-SUBGROUPS OF HYPERBOLIC GROUPS
cites work: The SQ-universality of hyperbolic groups
cites work: Q3202286
cites work: Subgroups of small Cancellation Groups
cites work: Endomorphisms of hyperbolic groups. I: The Hopf property
DOI: 10.1006/JABR.1999.8104
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description: scientific article; zbMATH DE number 1423886