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A topological equivalence relation for finitely presented groups

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 $π_1(X) \cong G$ and $π_1(Y) \cong H$, 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 $π_1(X) \cong G$ 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.