See this page by Coornaert for an introduction and references. In particular, finitely generated hyperbolic groups are always finitely presented (so you don't need to add that condition). Also, the referenced page suggests that your first question, about finite index torsion free subgroups, is open. I looked in various references for your related question, but didn't find anything.

Edit: Aha! I finally looked in the right place. Apparently your related question is also open.

See this page by Coornaert for an introduction and references. In particular, finitely generated hyperbolic groups are always finitely presented (so you don't need to add that condition). Also, the referenced page suggests that your first question, about finite index torsion free subgroups, is open. I looked in various references for your related question, but didn't find anything.

Edit: Aha! I finally looked in the right place. Apparently your related question is also open.

See this page by Coornaert for an introduction and references. In particular, finitely generated hyperbolic groups are always finitely presented (so you don't need to add that condition). Also, the referenced page suggests that your first question, about finite index torsion free subgroups, is open. I looked in various references for your related question, but didn't find anything.

See this page by Coornaert for an introduction and references. In particular, finitely generated hyperbolic groups are always finitely presented (so you don't need to add that condition). Also, the referenced page suggests that your first question, about finite index torsion free subgroups, is open. I looked in various references for your related question, but didn't find anything.

Edit: Aha! I finally looked in the right place. Apparently your related question is also open.