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:**


* Was it already constructed?
* Does it already has a name? Is there any closely related terminology?




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**P.S.**


* The group which I construct is in fact hyperbolic.
* The construction is simple, but it takes 2--3 pages. Let me know if you see a short way to do it.
* [Here](http://www.springerlink.com/content/k332884x7m10l654/), 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 ["telescopic" actions](https://arxiv.org/abs/1104.4814) now.



Universal group?
Аnswered to move the question to *answered status*.


We decided to use the term **telescopic action**.


Thank you all for your comments they were helpful for me and Dima.


