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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 decided to call it them "telescopic"; see Telescopic all-inclusive" actions now.
4 added 76 characters in body

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 decided to call it "telescopic".telescopic"; see Telescopic actions
3 added 69 characters in body

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: