  In this paper, we consider an equivalence relation within the class of
finitely presented discrete groups attending to their asymptotic topology
rather than their asymptotic geometry. More precisely, we say that two finitely
presented groups $G$ and $H$ are "proper $2$-equivalent" if there exist
(equivalently, for all) finite $2$-dimensional CW-complexes $X$ and $Y$, with
$\pi_1(X) \cong G$ and $\pi_1(Y) \cong H$, so that their universal covers
$\widetilde{X}$ and $\widetilde{Y}$ are proper $2$-equivalent. It follows that
this relation is coarser than the quasi-isometry relation. We point out that
finitely presented groups which are $1$-ended and semistable at infinity are
classified, up to proper $2$-equivalence, by their fundamental pro-group, and
we study the behaviour of this relation with respect to some of the main
constructions in combinatorial group theory. A (finer) similar equivalence
relation may also be considered for groups of type $F_n, n \geq 3$, which
captures more of the large-scale topology of the group. Finally, we pay special
attention to the class of those groups $G$ which admit a finite $2$-dimensional
CW-complex $X$ with $\pi_1(X) \cong G$ and whose universal cover
$\widetilde{X}$ has the proper homotopy type of a $3$-manifold. We show that if
such a group $G$ is $1$-ended and semistable at infinity then it is proper
$2$-equivalent to either ${\mathbb Z} \times {\mathbb Z} \times {\mathbb Z}$,
${\mathbb Z} \times {\mathbb Z}$ or ${\mathbb F}_2 \times {\mathbb Z}$ (here,
${\mathbb F}_2$ is the free group on two generators). As it turns out, this
applies in particular to any group $G$ fitting as the middle term of a short
exact sequence of infinite finitely presented groups, thus classifying such
group extensions up to proper $2$-equivalence.
