Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

If G is a discrete cofinite volume subgroup of PSL(2,C),then G acts on H3, H3/G is a 3-dim hyperbolic orbifold N with finite volume, my question is : Is it right in most situations that we can find a hyperbolic 3 manifold M as a finite covering space of N? This question is equivalent to the following : do most dicrete cofinte volume klein groups have a finit order torsion free subgroup?

share|improve this question

1 Answer 1

up vote 8 down vote accepted

Yes, this is true for all of them. Any finitely generated matrix group has a torsion-free subgroup of finite index; this is the so-called "Selberg's lemma". A canonical source is Ratcliffe's Hyperbolic Manifolds book (you can probably find the relevant section on google books for free, or on gigapedia.com if you are so inclined).

share|improve this answer
thanks very much! It's very useful for me. –  strygwyr Oct 2 '11 at 15:30
gigapedia is now library.nu –  Junyan Xu Nov 29 '11 at 13:29
gigapedia.com still works fine for me :) –  Igor Rivin Nov 29 '11 at 13:57

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.