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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 $\mathbb R^3$?

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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
up vote 10 down vote accepted

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.

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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.

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Thank you, Ian! I realized that I referred to [Scott-Tucker] in a preprint last October, but this was for a different example, and I did not look at other parts of the paper. I do not have electonic access, so I get the paper Monday. Incidentally, am I correct that the Tameness Conjecture is open for nonpositively curved 3-manifolds? – Igor Belegradek Feb 23 '13 at 0:42
Freedman was aware of Scott-Tucker, I'm not sure why it's not referenced in their paper (I learned of it in Freedman's seminar). The tameness conjecture is open for non-positively curved 3-manifolds, although the proof I gave works in the pinched negatively curved case, and I think with just a negative upper bound on curvature (I believe this extension appears in notes of Bowditch). – Ian Agol Feb 23 '13 at 5:26
Is "negative upper bound on curvature" really enough? I thought, in your paper "hyperbolic cusps" are assumed, while Bowditch gives a proof in the pinched negative case. – Igor Belegradek Feb 23 '13 at 12:27
Ok, I'm not sure about the case of only an upper bound on curvature, although I suspect the proof should carry through. – Ian Agol Feb 23 '13 at 19:41
I know little of dimension 3 but looking at Bowditch's notes I see potential difficulties starting with existence of thick/thin decomposition (Margulis lemma need a lower curvature bound), and the topology of cusps (which can get ugly). Bowditch also uses some results on convex hulls that depend on a paper of Michael Anderson for which lower curvature bound is needed. – Igor Belegradek Feb 23 '13 at 22:27

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