## Illinois Journal of Mathematics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 17 November 2009

https://projecteuclid.org/euclid.ijm/1258488167

Digital Object Identifier
doi:10.1215/ijm/1258488167

Mathematical Reviews number (MathSciNet)
MR2031335

Zentralblatt MATH identifier
1031.20017

#### Citation

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. https://projecteuclid.org/euclid.ijm/1258488167