Introduction

A group $G$ is said to be properly $3$ -realizable, P$3$ R for short, if there is a finite $2$ -complex $X$ with $\pi_1(X) \cong G$whose universal cover $\tilde{X}$ is proper homotopy equivalent to a $3$ -manifold (which may be called a proper $3$ -realization of $G$). One can only ensure $\tilde{X}$ always to be proper homotopy equivalent to some $4$ -manifold, see [CL05].
The class of P$3$ R-groups is known to contain the classes of finite groups, finitely generated abelian groups, direct products and ascending HNN-extensions of finitely presented groups, simply connected at $\infty$ groups and one-relator groups. Also, the class of P$3$ R-groups is closed under amalgamated products (and HNN-extensions) over finite groups. See [CL05,CLRQ] and related works.
It is worth noting P$3$ R-groups have $H^2(G,\mathbb{Z}G)$ free abelian. This is related to a long standing conjecture attributed to Hopf. Recent results, see [CLQ06], show that not all semistable groups are P$3$ R. It is unknown whether or not all P$3$ R-groups are semistable. This relates for $1$ -ended groups to the Covering Conjecture as follows. Clearly, the fundamental group $G$ of a closed $3$ -manifold is P$3$ R. A proper $3$ -realization of such $G$ is, up to proper homotopy equivalence, an open $3$ -manifold with a collection of $3$-balls removed. If there were non-semistable P$3$ R-groups this manifold would be a punctured $\mathbb{R}^3$ showing the conjecture, see [CLQ06].