ROY , R.

We recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with π1(X) ∼= G, the universal cover X has the proper homotopy type of a 3-manifold. This purely topological property is closely related to the asymptotic behavior of the group G. We show that proper 3-realizability is also a...

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...