Higher dimensions

Proper $3$ -realizability can be alternetively described as follows: a group $G$ of type $F_2$ for which the $2$ -skeleton of the universal cover of a $K(G,1)$ is proper homotopy equivalent to a $3$ -manifold. Similarly, we may say that a $F_n$ group $G$ is proper $2n-1$ -realizable, $P(2n-1)R$ for short, if the $n$ -skeleton of the universal cover of a $K(G,1)$ is properly homotopy equivalent to a $2n-1$ -manifold. As in the $3$ -dimensional case, the embedding theorems [J.R65],[DR93] and [CFLQ03] only ensure proper homotopy equivalence to $2n$ -manifolds. We ask ourselves which kind of algebraic properties consecuences has this property on $G$ and when, for which $n$ , this property becomes trivial.