Illinois Journal of Mathematics

The residual finiteness of ascending HNN-extensions of certain soluble groups

A. H. Rhemtulla and M. Shirvani

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If $G$ is a group with an injective endomorphism $\phi$, then the HNN-extension $G_\phi = \langle G, t \,:\, t^{-1}gt=g \phi \,\, \mbox{for all}\,\, g \in G\rangle $ is called the ascending HNN-extension of $G$ determined by $\phi$. We prove that $G_\phi$ is residually finite when $G$ is either finitely generated abelian-by-polycyclic-by-finite or reduced soluble-by-finite minimax. We also provide an example of a $3$-generator residually finite soluble group $G$ of derived length $3$ with a non-residually-finite ascending HNN-extension.

Article information

Illinois J. Math., Volume 47, Number 1-2 (2003), 477-484.

First available in Project Euclid: 17 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E26: Residual properties and generalizations; residually finite groups
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 20F16: Solvable groups, supersolvable groups [See also 20D10]


Rhemtulla, A. H.; Shirvani, M. The residual finiteness of ascending HNN-extensions of certain soluble groups. Illinois J. Math. 47 (2003), no. 1-2, 477--484. doi:10.1215/ijm/1258488167.

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