# Non-tame 3-manifolds covered by the Euclidean space

An open 3-manifold is tame if it is homeomorphic to the interior of a compact manifold. Is there a (known) example of an open 3-manifold that is not tame, has finitely generated fundamental group and universal cover homeomorphic to $\mathbb R^3$?

-
I would be interested in an example of an atoroidal, irreducible 3-manifold such that every cover with finitely generated fundamental group is tame, but which does not admit a hyperbolic metric. –  Ian Agol Feb 21 '13 at 0:33
What does atoroidal mean for a non-tame manifold? I am guessing that the manifold has finitely many tame ends of zero Euler characteristic and any essential torus/Klein bottle comes from one of these ends, is this correct? –  Igor Belegradek Feb 21 '13 at 0:57
Yes, if you mean essential immersed torus (it is the same hypothesis as in the geometrization theorem) –  Ian Agol Feb 21 '13 at 3:10
Neat question, Ian. –  Richard Kent Feb 21 '13 at 3:15

Yes, see the following paper of Freedman and Gabai for lots of examples:

Freedman and Gabai, Covering a nontaming knot by the unlink. Algebr. Geom. Topol. 7 (2007), 1561–1578.

-
Thanks! Their paper also answers another question I was wondering about: it gives a 3-manifold with universal cover $\mathbb R^3$ and $\mathbb Z\ast\mathbb Z$ fundamental group that is not hyperbolizable. I was not aware this is possible. –  Igor Belegradek Feb 20 '13 at 23:33
I meant to say "not hyperbolizable (by the solution of the Tameness Conjecture)". –  Igor Belegradek Feb 20 '13 at 23:47
Glad to be of service, Igor! –  Richard Kent Feb 21 '13 at 3:14
I believe the original such examples (with fundamental group $\mathbb{Z}$) were due to Scott and Tucker.