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 G with infinite monodromy, if G G is CAT(0) then the monodromy representation is injective. We extend this to a more general result: Let G G be a group with a normal surface subgroup R R . Assume G / R G/R satisfies the property that for every infinite normal subgroup Λ \Lambda of G / R G/R , there is an infinite finitely generated subgroup Λ 0 > Λ \Lambda _0>\Lambda so that the centralizer C G / R ( Λ 0 ) C_{G/R}(\Lambda _0) is finite. We then prove that if G G is CAT(0) with infinite monodromy, then the monodromy representation has a finite kernel. This applies in particular if G / R G/R is acylindrically hyperbolic.