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?

----

**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.