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.
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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.