# Hyperbolic manifold of dim 3 with finite volume.

The geometrization Theorem for 3-manifolds classifies all oriented compact (without boundary ) 3-manifolds. Is there a theorem which classifies all hyperbolic oriented 3-manifold (without boundary ) with finite volume ?

• The geometrization theorem yields a decomposition theorem into pieces, but does not provide a classification of the pieces (by the way, it holds not only for closed manifolds). Part of the classification is to classify the pieces and in the hyperbolic case this is an issue (In the hyperbolic case, the solution to the virtual Kaken conjecture is a great improvement although it does not mean exactly a classification)
– YCor
May 25, 2015 at 10:20
• By the way I don't understand if you're speaking of 3-folds or hyperbolic 3-folds, so I may misunderstand your question. The geometrization theorem is a decomposition theorem for 3-folds, which for a geometrizable 3-fold such as a hyperbolic 3-fold is a tautology.
– YCor
May 25, 2015 at 10:22
• @YCor What I'am asking is the classification of subgroups of $PSL(2,C)$ which act nicely on $\mathbf{H}^{3}$ i.e. the quotient is a hyperbolic 3-manifold with a finite volume. What is known about this classification ? May 25, 2015 at 12:33
• this was my guess, but you could reformulate your first sentence as something "By work of Agol, Wise and others, the Virtual Haken theorem provides a classification of closed hyperbolic 3-manifolds (up to finite covering, and modulo a suitable classification of pseudo-anosov elements in the mapping class groups)."
– YCor
May 25, 2015 at 14:41