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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?

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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 if you are so inclined).

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

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