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


I believe the original such examples (with fundamental group $\mathbb{Z}$) were due to Scott and Tucker. 

