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I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can provide me some insight.

Let $G$ be a group with an injective endomorphism $\phi$, then the HNN-extension $$G_\phi = \left<G,t \mid t^{-1} gt= \phi(g) \right> $$ is called the ascending HNN-extension of $G$ determined by $\phi$.

Questions:

Does there exist a finitely generated, torsion group $G$, with an injective but non-surjective endomorphism $\phi$, such that $G_\phi$ is residually finite?

Thoughts: Clearly, $G$ needs to be residually finite. In this paper Sapir & Wise showed that an ascending HNN extension of a finitely generated residually finite torsion group need not be residually finite. I was wondering, given the fact that $G_\phi$ is residually finite, can $G$ be a torsion group?

I tried to show that such a group does not exist. This paper contains the criterion for the HNN extension to be residually finite, but I can't seem to get a contradiction.

Any reference/examples would be really appreciated, thanks in advance.

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  • $\begingroup$ I'm sure you're also aware of the theorem of Borisov and Sapir, that every ascending HNN extension of finitely generated linear groups is residually finite (arxiv.org/abs/math/0309121). Of course, that doesn't help with your question, but it might contain some useful ideas. By the way, I'm not surprised you can't derive a contradiction: I would certainly guess that such a group does exist. For example, it seems conceivable that Lysenok's finitely presented extension of Grigorchuk's group could be residually finite. $\endgroup$
    – HJRW
    Commented Aug 12, 2022 at 14:38
  • $\begingroup$ @HJRW, Sapir says in his 2009 Groups St, Andrew paper that Lysenok's extension isn't RF. $\endgroup$ Commented Aug 12, 2022 at 15:09
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    $\begingroup$ @HJRW He kind of gives a reason. He says to "see [SW]", which is the paper by Sapir & wise in J. Pure Appl. Algebra from 2002 ("Ascending HNN extensions..."). In there, it is mentioned that "...there exists an ascending HNN extension of $G$ [Grigorchuk's group] which is not residually finite, and the proof of this substantially follows the lines of the proof of Lemma 2.1." A proof is then given that an HNN-extension of $G$ has only metabelian proper hom. images, so not RF; I don't know if that HNN-extension is Lysënok's (I'm sure you can see it quicker than me). This is on pp. 194--195. $\endgroup$ Commented Aug 12, 2022 at 17:40
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    $\begingroup$ @HJRW, didn't Olshanskii and Sapir have a torsion by cyclic nonamenable group that is finitely presented to get the first finitely presented nonamenable group without free subgroups. $\endgroup$ Commented Aug 12, 2022 at 23:19
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    $\begingroup$ @HJRW An fp ascending HNN extension of Grigorchuk's group is considered in [R. I. Grigorchuk. An example of a finitely presented amenable group that does not belong to the class EG. Sb. Math. 189 (1998), no. 1-2, 75–95]. I don't know if it coincides with Lysënok's, but probably it does. It's due to Sapir-Wise 2002 that it's just-non-metabelian and hence isolated as marked group (and hence can't be RF) — this is mentioned in §5.7 in Cornulier-Guyot-Pitsch 2007). $\endgroup$
    – YCor
    Commented Aug 16, 2022 at 4:46

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