Surface-by-free hyperbolic groups are cubulable when the monodromy comes from a sufficiently thick tight tree of homologous curves, because the group contains an essential incompressible quasiconvex track.
Residually finite rationally solvable groups and virtual fibring
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abstract
We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.
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Cubulating Surface-by-free Groups
Surface-by-free hyperbolic groups are cubulable when the monodromy comes from a sufficiently thick tight tree of homologous curves, because the group contains an essential incompressible quasiconvex track.