# A topological equivalence relation for finitely presented groups

@article{Cardenas2020ATE, title={A topological equivalence relation for finitely presented groups}, author={M. C'ardenas and Francisco F. Lasheras and Antonio Quintero and Ranja Roy}, journal={arXiv: Geometric Topology}, year={2020} }

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…

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