Homology growth and (non)fibering

Suppose a group \(G\) has a finite \(K(G,1)\) space \(X\), and suppose we have a sequence of deeper and deeper regular finite sheeted covers of \(X\), so that the corresponding sequence of normal subgroups intersect at \(\{1\}\).  What can we say about homology of these covers? Rationally, the answer is given by the celebrated Lück Approximation theorem: the normalized Betti numbers of the covers limit to the \(\ell^2\)-Betti numbers of \(G\).

I will discuss this and the corresponding notions for torsion part of homology. I will also explain recent computations for right-angled Artin groups and their relatives, and how they can be used to construct a closed aspherical Gromov hyperbolic 7-manifold which does not virtually fiber over the circle.  This is based on a joint work joint with Grigori Avramidi and Kevin Schreve.