show/hide this revision's text 5 deleted 17 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:

  • 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.
  • 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.
show/hide this revision's text 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:

  • 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.
  • 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
show/hide this revision's text 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:

  • 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.
  • 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".
show/hide this revision's text 2 added 180 characters in body
show/hide this revision's text 1