  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 $\Lambda$ of $G/R$, there
is an infinite subgroup $\Lambda_0<\Lambda$ so that the centralizer
$C_{G/R}(\Lambda_0)$ 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.
