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.
 - My construction is simple, but it takes 2--3 pages. Let me know if you see a short way to do it.
 -  [Here][1], the term "universal group" was used in very similar context (thanks to D. Panov for the reference).
 - Thanks to all your comments, we decided to call it "telescopic"; see [Telescopic actions][2]


 


  [1]: http://www.springerlink.com/content/k332884x7m10l654/
  [2]: http://front.math.ucdavis.edu/1104.4814