Conjugacy separability and outer automorphism groups of certain HNN extensions.
type: Item
instance of: scholarly article
title: Conjugacy separability and outer automorphism groups of certain HNN extensions.
zbMATH Open document ID: 1239.20034
author: K. B. Wong
author: P. C. Wong
published in: Journal of Algebra
publication date: 2011-12-19T00:00:00Z
review text: A group \(G\) is conjugacy separable if for every \(x\in G\) its conjugacy class \(\{x\}^G\) is closed with respect the profinite topology in \(G\). Evidently conjugacy separability implies residual finiteness. It is well known that the automorphism group of a finitely generated residually finite group is residually finite. However [see \textit{D. T. Wise}, Bull. Lond. Math. Soc. 35, No. 1, 23-29 (2003; Zbl 1027.20014)], there exist finitely generated residually finite groups where the outer automorphism group is not residually finite. On the other hand \textit{E. K. Grossman} [in J. Lond. Math. Soc., II. Ser. 9, 160-164 (1974; Zbl 0292.20032)] proved that if a finitely generated group is conjugacy separable and every conjugating automorphism is also an inner automorphism then its outer automorphism group is residually finite. In this paper, on one hand, the authors study the conjugacy separability of certain HNN-extensions with associated subgroups in the center of the base group and obtain: Theorem: Let \(G=\langle t,A\mid t^{-1}Ht=K,\;\varphi\rangle\) be the HNN-extension of the polycyclic-by-finite group \(A\) and proper associated subgroups \(H\) and \(K\) in the center of \(A\). The group \(G\) is conjugacy separable if and only if it is residually finite. This theorem is inspired by an analogous theorem of \textit{E. Raptis, O. Talelli} and \textit{D. Varsos} [J. Algebra 199, No. 1, 327-336 (1998; Zbl 0891.20023)] where the base group is a finitely generated Abelian group and generalizes the results there. On the other hand they prove: Theorem: Let \(G=\langle t,A\mid t^{-1}Ht=K,\;\varphi\rangle\) be the HNN-extension of the group \(A\) and proper associated subgroups \(H\) and \(K\) in the center of \(A\). Then every conjugating endomorphism of \(G\) is an inner automorphism. Combing this result with the Grossman's result it is obtained: Theorem: Let \(G=\langle t,A\mid t^{-1}Ht=K,\;\varphi\rangle\) be the HNN-extension of the polycyclic-by-finite group \(A\…
reviewed by: Dimitrios Varsos
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 5990145
zbMATH Keywords: outer automorphisms
zbMATH Keywords: residually finite groups
zbMATH Keywords: conjugacy separable groups
zbMATH Keywords: HNN-extensions
zbMATH Keywords: polycyclic-by-finite groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.02.038
OpenAlex ID: W1964269425
cites work: Residual finiteness of outer automorphism groups of certain pinched 1-relator groups
cites work: RESIDUAL FINITENESS OF OUTER AUTOMORPHISM GROUPS OF FINITELY GENERATED NON-TRIANGLE FUCHSIAN GROUPS
cites work: A characterization of residually finite HNN-extensions of finitely generated Abelian groups
cites work: Automorphism Groups of Residually Finite Groups
cites work: Some two-generator one-relator non-Hopfian groups
cites work: Conjugacy in Nilpotent Groups
cites work: OUTER AUTOMORPHISM GROUPS OF CERTAIN TREE PRODUCTS OF ABELIAN GROUPS
cites work: Recursively enumerable degress and the conjugacy problem
cites work: Q3866296
cites work: Q3972206
cites work: Conjugate separability in polycyclic groups
cites work: On the Residual Finiteness of Certain Mapping Class Groups
cites work: OUTER AUTOMORPHISM GROUPS OF CERTAIN POLYGONAL PRODUCTS OF GROUPS
cites work: Conjugacy separability of HNN-extensions of abelian groups
cites work: A criterion for the conjugacy separability of certain HNN extensions of groups
cites work: OUTER AUTOMORPHISM GROUPS OF POLYGONAL PRODUCTS OF CERTAIN CONJUGACY SEPARABLE GROUPS
cites work: Q4145882
cites work: On the conjugacy separability of certain graphs of groups
cites work: Groups that are residually finite with respect to conjugacy
cites work: A RESIDUALLY FINITE VERSION OF RIPS'S CONSTRUCTION
cites work: Conjugacy Separability of Certain HNN Extensions of Conjugacy-Separable Groups
cites work: THE RESIDUAL FINITENESS OF CERTAIN HNN EXTENSIONS
cites work: Residual finiteness of outer automorphism groups of certain tree products
DOI: 10.1016/J.JALGEBRA.2011.02.038
Recommended article: Conjugacy Separability of Certain HNN Extensions of Conjugacy-Separable Groups
Recommended article: Class-preserving automorphisms of certain HNN extensions.
Recommended article: Class-preserving automorphisms of certain HNN extensions
Recommended article: THE RESIDUAL FINITENESS OF CERTAIN HNN EXTENSIONS
Recommended article: Subgroup separability and conjugacy separability of certain HNN extensions.
Recommended article: Ascending HNN extensions of polycyclic groups are residually finite.
Recommended article: Residual finiteness of outer automorphism groups of certain tree products
Recommended article: Cyclic subgroup separability of certain HNN extensions of finitely generated Abelian groups
Recommended article: Subgroup separability of certain HNN extensions of finitely generated Abelian groups
Recommended article: Residual finiteness, subgroup separability and conjugacy separability of certain HNN extensions
description: scientific article; zbMATH DE number 5990145