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

TL;DR

This paper investigates conditions under which surface group extensions are CAT(0), extending previous results to more general groups with specific properties, and shows that acylindrically hyperbolic groups satisfy these conditions.

Contribution

It generalizes the necessary condition for surface-by-surface groups to be CAT(0) to a broader class of groups with certain subgroup properties.

Findings

If a group with a normal surface subgroup is CAT(0), then the monodromy has finite kernel.

Acylindrically hyperbolic groups satisfy the subgroup property needed for the main result.

The work extends the understanding of geometric structures on surface group extensions.

Abstract

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.