$L^2$ Betti numbers and kernels of maps to $\mathbb Z$

In 2018, Kielak proved that if $G$ is a finitely generated RFRS group, then $G$ virtually algebraically fibers if and only if its first $L^2$-Betti number is zero. We will present a generalisation of this result to higher degrees and to positive characteristic and discuss connections to mod $p$ homology growth. We will also see that if the top-degree $L^2$-Betti number of $G$ vanishes, then $G$ admits a virtual map to $\mathbb Z$ with kernel of cohomological dimension strictly less than that of $G$. In particular, if $G$ is 2-dimensional and has vanishing second $L^2$-Betti number, then $G$ is virtually free-by-cyclic and therefore coherent.