3
$\begingroup$

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, what could one say about $\{e\}\leq H$ being benign? and what about $H\leq H$? I had some ideas along that line but nothing "ripe" enough...
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.

edit: I've copied this question over to MO in the hope of getting some good thinking directions.

$\endgroup$
  • $\begingroup$ (1) $\;\{e\}\;$ is benign in $\;H\;$ iff $\;H\;$ can be embedded in a f.p. group, and (2) $\;H\;$ is benign in itself iff $\;tht^{-1}=h\;\forall h\in H (\iff \langle t\rangle \le C_{H_H})\;$ is embeddable in a f.p. group $\endgroup$ – Timbuc Feb 5 '15 at 9:49
  • $\begingroup$ @Timbuc First of all-TY! I had gotten that much while doodling with the definitions, but I'm still lacking the insight I was hoping to gain...Do you have an idea how to tranlate this into something more intuitive? I don't need anything very formal or rigorous... $\endgroup$ – ShlomiF Feb 5 '15 at 10:17
  • $\begingroup$ Not really, sorry. This concept is a rather subtle one, and I'd love to see one single example of a non-benign subgroup of a group...yet nobody using the concept, beginning by Higman himself in his paper, bothers to give one darn such example, as far as I can see. Solution? Approach someone from group theory in your school or else write somebody who's been dealing with this asking him/her about this. The following is a paper with an "easy" proof of Higman's Embedding theorem: cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/… $\endgroup$ – Timbuc Feb 5 '15 at 10:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.