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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?

edited Oct 2 2011 at 17:18

Qfwfq
8,481●34●86

asked Oct 2 2011 at 14:22

strygwyr
145●4

Igor Rivin · Accepted Answer · 2011-10-02 15:12:40Z UTC

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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).

answered Oct 2 2011 at 15:12

Igor Rivin
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	thanks very much! It's very useful for me. – strygwyr Oct 2 2011 at 15:30
	gigapedia is now library.nu – Junyan Xu Nov 29 2011 at 13:29
	gigapedia.com still works fine for me :) – Igor Rivin Nov 29 2011 at 13:57

Can most 3 dimensional hyperbolic orbifolds with finite volume be covered by a hyperbolic manifold?

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