On the Bass-Lubotzky question about quotients of hyperbolic groups
type: Item
instance of: scholarly article
title: On the Bass-Lubotzky question about quotients of hyperbolic groups
zbMATH Open document ID: 0957.20026
published in: Journal of Algebra
publication date: 2001-01-10T00:00:00Z
review text: The author proves the following Theorem 1. Every non-elementary hyperbolic group \(G\) has a non-trivial quotient \(Q=G/N\) which is finitely presented and which possesses no non-trivial finite homomorphic images. The result was asked by Bass and Lubotzky, who needed it to complete their construction of counter-examples to the Platonov conjecture. The proof of Theorem 1 uses techniques of generalized small cancellation which were developed in previous papers by the author. The author proves also Theorem 2. The following conditions are equivalent: (i) There exists a non-residually finite hyperbolic group. (ii) There exists a non-trivial hyperbolic group having no proper subgroups of finite indices. (iii) There exists a finite collection of non-elementary hyperbolic groups such that the set of the isomorphism classes of their common non-Abelian, finite, simple homomorphic images is finite. Whether statements (i), (ii) and (iii) are true is an open problem. In his paper [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], \textit{M. Gromov} conjectures that statement (ii) is true.
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F06
zbMATH DE Number: 1448612
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: finitely presented groups
zbMATH Keywords: Gromov hyperbolicity
zbMATH Keywords: small cancellation conditions
zbMATH Keywords: finite homomorphic images
author: Alexander Y. Olshanskii
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: jabr.1999.8170
OpenAlex ID: W2012644333
cites work: Nonarithmetic superrigid groups: Counterexamples to Platonov's conjecture
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DOI: 10.1006/JABR.1999.8170
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description: scientific article; zbMATH DE number 1448612