Iterated Kodaira Fibrations and Surface Bundles

In order to determine when surface-by-surface bundles are non-positively curved, Llosa Isenrich and Py give a necessary condition: given a surface-by-surface group G with infinite monodromy, if G is CAT(0) then the monodromy representation is injective. We extend this to a more general result: Let G be a group with a normal surface subgroup R. Assume G/R satisfies the property that for every infinite normal subgroup Λ of G/R, there is an infinite subgroup Λ0<Λ so that the centralizer CG/R(Λ0) is finite. If G is CAT(0) with infinite monodromy, then the monodromy representation has a finite kernel. We prove that acylindrically hyperbolic groups satisfy this property.