How different is the universal cover of a given finite 2-complex from a
3-manifold (from the proper homotopy viewpoint)? Regarding this question, we
recall that a finitely presented group $G$ is said to be properly 3-realizable
if there exists a compact 2-polyhedron $K$ with $\pi_1(K) \cong G$ whose
universal cover $\tilde{K}$ has the proper homotopy type of a PL 3-manifold
(with boundary). In this paper, we study the asymptotic behavior of finitely
generated one-relator groups and show that those having finitely many ends are
properly 3-realizable, by describing what the fundamental pro-group looks like,
showing a property of one-relator groups which is stronger than the QSF
property of Brick (from the proper homotopy viewpoint) and giving an
alternative proof of the fact that one-relator groups are semistable at
infinity.