Aspherical manifold with superperfect fundamental group and non-trivial center? I am interested in knowing if there is a closed, (smooth) aspherical manifold $M$ (hyperbolic would be best) with superperfect fundamental group (that is to say, with $H_1(\pi_1(M);\mathbb{Z}) = H_2(\pi_1(M);\mathbb{Z}) = 0$; note $H_1(\pi_1(M);\mathbb{Z}) = 0$ is equivalent to perfect) and non-trivial center ($\mathbb{Z}_2$ would be best, but any f.g. Abelian group will do).
Also, assuming there is a manifold that fits the criteria, I would likely need a handlebody decomposition for the manifold, assuming the "standard handlebody procedure" for producing a closed, smooth manifold from a prescribed finite presentation of a/the fundamental group does not yield the smooth manifold in question (e.g., the "standard manifold" is not aspherical).
I found many hyperbolic 3-manifolds with superperfect fundamental group using SnapPy, but SnapPy evidently doesn't have a center "method" for the fundamental group method/class attached to 3-manifolds. Sage/GAP/MAGMA also appear not to be able to compute the center for an infinite finitely-presented fundamental group.
Thanks much in advance. I realize this is kind-of "shooting for the moon/stars".
 A: Per Lee's request, I've turned the discussion into an answer. 
Suppose that $M$ is a closed connected oriented hyperbolic $n$ manifold, for $n$ at least two.  Then the fundamental group contains no torsion elements; also its center is trivial.  Both claims are exercises from the classification of isometries of hyperbolic $n$-space. 
Thus there are no hyperbolic homology three-spheres with the property you desire.  
However, there are many three-manifolds that are (integral) homology spheres (and so $\pi_1$ is super-perfect) and where $\pi_1$ has non-trivial abelian centre.  These are found among the Seifert fibered spaces.  The most famous of these is the Poincare homology sphere, but this example is ruled out by your requirement that $M$ be aspherical.
More specifically, you should consider the "aspherical Brieskorn homology spheres" $\Sigma(p, q, r)$.  These are described at the Wikipedia page linked to immediately above, which also briefly sketches a presentation of their fundamental group.  If you want Snappy to compute things for you, then consult Figure 1 of this paper (say) for a surgery description of $\Sigma(p, q, r)$.  This paper also gives a short but useful discussion of the fundamental group. 
