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

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. [ABSTRACT FROM AUTHOR]