This article is cited in 3 papers

On the Residual Finiteness of Descending HNN-Extensions of Groups

D. N. Azarov

Ivanovo State University

Abstract: Let $G$ be a group of finite generic rank, $\varphi $ an injective endomorphism of the group $G$, and $G(\varphi)$ the descending HNN-extension of $G$ corresponding to the endomorphism $\varphi$. Let the index of the subgroup $G\varphi$ in $G$ be finite and equal to $n$. It is proved that, if the group $G$ is almost residually $\pi$-finite for some set $\pi$ of primes coprime to $n$, then the group $G(\varphi)$ is residually finite. This generalizes a series of known results, including the Wise–Hsu theorem on the residual finiteness of an arbitrary descending HNN-extension of any almost polycyclic group.

Keywords: residual finiteness, descending HNN-extension, almost residually $\pi$-finite group.

UDC: 512.543

Received: 28.12.2011
Revised: 06.01.2014

DOI: 10.4213/mzm9312

 English version: Mathematical Notes, 2014, 96:2, 161–165

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