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Disclaimer! This is a copy of a question I posted on M.SE! I still think the question belongs there but I'm not getting any answers so I'm dublicating with slight changes:
I've been studying a proof of Higmann's Embedding Theorem which makes use of the notion of a "benign subgroup".
The definition is quite straight-forward:
$G\leq H \ is\ benign\ in\ H \Leftrightarrow$ $H_G := \langle H, t; tgt^{-1}=g\ \forall g\in G\rangle\ is\ embeddable\ in\ a\ f.p\ group$
I've been trying to get an intuitive understanding of this construction but haven't been able to come up with anything...
Some of the directions I've looked at are:
$H_G$ is obviously an HNN-extension of H, with G being trivialy isomorphic to itself. So I thought there might be general theorems about HNN-extensions which would give me a better understanding...but couldn't find any that did.
I've tried to figure out when certain subgroups would be benign or not: for instance, the trivial subgroup is benign in $H$ if $H$ is already embedable in a f.p group, and $H$ is benign in itself means the stable letter is in the center of the HNN-extension.
As was pointed out on MSE it would also be good to see an example of a non-benign subgroup, but this would probably involve looking at groups which aren't recursively presented...
Also, one might try use the the meaning of benign to get a clue.
I'm feeling rather stupid to say the least because it seems to be at the tip of my tongue and on the brink of my understanding, and I'll be very gratefull for some insight.
Thanx a bunch,
Shlomi.

I don't necessarily need anything very rigorous...just the apropriate hand-waving :-)

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    $\begingroup$ Once Higman's theorem is known, the notion of benign subgroup just means that $H$ admits a generating sequence $(u_n)$ over which it is recursively presentable, and that $G$ can be generated by some sequence of words in the $(u_n)$ that is computable. So it rather appears as a provisional definition inside the proof of Higman's theorem. $\endgroup$
    – YCor
    Commented Feb 5, 2015 at 19:17
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    $\begingroup$ An example of a non-benign subgroup in the free group $\langle x,y\rangle$ is the subgroup (freely) generated by the $x^nyx^{-n}$ when $n$ ranges over some non-recursively-enumerable subset of $\mathbf{Z}$. $\endgroup$
    – YCor
    Commented Feb 5, 2015 at 19:19
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    $\begingroup$ @StefanKohl it was asked by me, and I specificaly noted that at the top of my question. I didn't get any real good answers or comments there so I posted here as recommended by a meta i can't find now. Do as you wish by I'm pretty sure that I'm actualy playing fair game. I also updated my question on MSE to link to this question... $\endgroup$
    – ShlomiF
    Commented Feb 5, 2015 at 20:04
  • $\begingroup$ @YCor I was trying to build something like your example but you obviously think faster :-). Your first comment is something I'll need to think about some more though...Thank you very much! $\endgroup$
    – ShlomiF
    Commented Feb 5, 2015 at 20:06
  • $\begingroup$ @StefanKohl, that is a little harsh, give the disclaimer... $\endgroup$
    – HJRW
    Commented Feb 6, 2015 at 9:43

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