Existence of finite index torsion-free subgroups of hyperbolic groups Question. Is it true that each infinite hyperbolic group
has a torsion-free subgroup of finite index?
Are there counterexamples, or positive results for some large subclasses of hyperbolic groups?
For example, is the answer positive for orbifold fundamental groups of negatively curved orbifolds? More precisely, I am interested the most in the case of orbifolds with locally $CAT(0)$ metric. I guess it will be hard to construct a counterexample in this category.
Related question. Is it known that every nontrivial hyperbolic group has a proper subgroup of finite index?
Just to recall, a definition of hyperbolic group is here  https://en.wikipedia.org/wiki/Hyperbolic_group .
Added. Note, that every hyperbolic group is finitely presented (thanks to Sam Nead).
 A: 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.
A: This is a well known open problem. The following properties are equivalent 
a) Every hyperbolic group is residualy finite 
b) Every hyperbolic group has a finite index torsion-free subgroup. 
The proof is either here: Olʹshanskiĭ, A. Yu.
On the Bass-Lubotzky question about quotients of hyperbolic groups. 
J. Algebra 226 (2000), no. 2, 807--817 or here: Kapovich, Ilya; Wise, Daniel T. The equivalence of some residual properties of word-hyperbolic groups.  J. Algebra  223  (2000),  no. 2, 562--583 or can be given by exactly the same methods as in these two papers (I do not remember exactly which of these three possibilities hold). 
