Conditions on the monodromy for a surface group extension to be CAT(0)

Abstract:

In order to determine when surface-by-surface bundles are non-positively curved, Llosa Isenrich and Py [Math. Ann. 380 (2021), pp. 449–485] 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 finitely generated subgroup $\Lambda _0<\Lambda$ so that the centralizer $C_{G/R}(\Lambda _0)$ is finite. We then prove that if $G$ is CAT(0) with infinite monodromy, then the monodromy representation has a finite kernel. This applies in particular if $G/R$ is acylindrically hyperbolic.