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 hyperbolic group has a non-trivial subgroup of finite index?
Just to recall, a definition of hyperbolic group is here http://en.wikipedia.org/wiki/Hyperbolic_group .
Added. Note, that every hyperbolic group is finitely presented (thanks to Sam Nead).
Best Answer
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).