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 in [LIP21] give a necessary condition: given a surfaceby-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 Λ 0 < Λ so that the centralizer C G R (Λ 0 ) is finite. If G is CAT(0) with infinite monodromy, then the monodromy representation has finite kernel. We prove that acylindrically hyperbolic groups satisfy this property.