# When are isometry groups of hyperbolic 3-manifolds finite?

If $M$ is a finite volume hyperbolic 3-manifold, then its isometry group is finite. I believe this is also true for geometrically finite 3-manifolds. What is the most general condition on a hyperbolic 3-manifold that is known to ensure that its isometry group is finite?

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First, assume the holonomy group of your manifold is nonelementary and that the manifold is complete. Now the geom finite statement holds. Second, assume that fundamental group is finitely generated and the manifold is geom infinite. Then Thurston's covering theorem gives complete answer. –  Misha Mar 22 '13 at 2:10