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## Universal group?

I can construct a finitely presented group $G$ with the following property (which I use to construct something else).

Given a finitely preseted group $\Gamma$, there is a subgroup $G'\le G$ of finite index such that $$\Gamma=G'/\langle\mathrm{Tor}\, G'\rangle ,$$ where $\mathrm{Tor}\, G'\subset G'$ is the set of all elements of finite order.

I think to call such group $G$ universal.

Questions:

• Does it already has a name? Is there any closely related terminology?

P.S.

• The group which I construct is in fact hyperbolic.
• My construction is simple, but it takes 2--3 pages. Let me know if you see a short way to do it.
• Here, the term "universal group" was used in very similar context (thanks to D. Panov for the reference).
• Thanks to all your comments, we call them "all-inclusive" actions now.
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I advise against the word "universal", without more context at least. Call it Anton-universal or the Petrunin-Swiss-Army Group, or some useful modification of some synonym for "universal". Gerhard "Ask Me About System Design" Paseman, 2010.03.30 – Gerhard Paseman Mar 30 2010 at 23:48
Well, Swiss-Army Group is a nice name. But why not universal? --- after quick search I did not see that term "universal group" is used... – Anton Petrunin Mar 31 2010 at 0:34
The closest condition I've heard of is "SQ-universal": en.wikipedia.org/wiki/SQ_universal_group Your group satisfies a very strong form of "SQ-universal in the class of finitely presented groups". – Ian Agol Mar 31 2010 at 1:33
This property seems far too specific to be called simply "universal". I'd go with something like "TQ-universal". If you want to know whether someone else has done this, I'd try looking at the work of Olshanskii and his students. – HW Mar 31 2010 at 1:41
Anton, by a "universal finitely presented group" one usually means a finitely presented group that contains each finitely presented group as a subgroup. Such groups can be constructed via Higman's embedding theorem. If $Q$ is such a group, it is possible to cook up a hyperbolic group $G$ such that $Q$ is a quotient of $G$, and the kernel is normally generated by elements of finite order. This is of course not the same as what you do. – Igor Belegradek Mar 31 2010 at 3:07