On discrete solvgroups and Poénaru's condition (Q1291130)

\textit{V. Poénaru} [Topology 33, No, 1, 181-196 (1994; Zbl 0830.57010)] proved that, for a manifold \(M\) whose fundamental group \(G\) satisfies the condition \(P(2)\) below, the universal covering \(\widetilde M\) is simply connected at infinity (hence homeomorphic to \(\mathbb{R}^3\) if irreducible). A finitely generated group \(G\) is said to verify Poénaru's condition \(P(n)\) (for \(n\geq 2)\) if there exist a system \(B\) of generators and a function \(f:\mathbb{Z}_+\to \mathbb{R}_+\) fulfilling: (1) for all constants \(c>0\) we have: \(\lim_{k\to\infty} [k-c \cdot f(k)]=0\), (2) any two elements \(x,y\) which are sitting on the sphere of radius \(k\) and center 1 of the Cayley graph \(\Gamma(G,B)\) (endowed with the natural distance function \(d)\) and which are at distance at most \(n\), can be joined by a path inside the ball of radius \(k\) of \(\Gamma(G,B)\), whose length is less than \(f(k)\). By a simple example the author shows that Poénaru's condition \(P(2)\) still is not necessary for the homeomorphism \(\widetilde M^3\) to \(\mathbb{R}^3\). Namely, he shows that the class \[ G_A=(a,b,t\mid ab=ba,\;tat^{-1} =a^{A_{11}} b^{A_{12}},\;tbt^{-1}= a^{A_{21}} b^{A_{22}}) \] with an integer 2-by-2 hyperbolic matrix \(A\), \(\det A=1\), of solv-groups does not satisfy \(P(2)\) with respect to \((a,b,t)\).