Quasi-isometries and proper homotopy: The quasi-isometry invariance of proper 3-realizability of groups

Abstract

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 geometric property meaning that it is a quasi-isometry invariant for finitely presented groups. In fact, in this paper we prove that (after taking wedge with a single n-sphere) any two infinite quasiisometric groups of type Fn (n ≥ 2) have universal covers whose n-skeleta are proper homotopy equivalent. Recall that a group G is of type Fn if it admits a K(G, 1)-complex with finite n-skeleton.