Geodesic
On the Residual Finiteness of Descending HNN-Extensions of Groups
Matematičeskie zametki, Tome 96 (2014) no. 2, pp. 163-169

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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. 
@article{MZM_2014_96_2_a0,
     author = {D. N. Azarov},
     title = {On the {Residual} {Finiteness} of {Descending} {HNN-Extensions} of {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {163--169},
     publisher = {mathdoc},
     volume = {96},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_96_2_a0/}
}
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D. N. Azarov. On the Residual Finiteness of Descending HNN-Extensions of Groups. Matematičeskie zametki, Tome 96 (2014) no. 2, pp. 163-169. http://geodesic.mathdoc.fr/item/MZM_2014_96_2_a0/