Virtually RFRS Mapping Tori and Coherence

Abstract

Let $G$ be a finitely presented group that can be written as an extension \[ 1 \longrightarrow K \longrightarrow G \longrightarrow F_2 \longrightarrow 1 \] where $K$ is either the finitely generated free group $F_n$, $n > 2$ or the fundamental group of a closed surface of genus $g > 1$. We prove that if the image of the monodromy map $\rho \colon F_2 \to \operatorname{Out(K)}$ contains an element $\varphi \in \operatorname{Out(K)}$ such that the mapping torus $K \rtimes_{\varphi} \Bbb{Z}$ is virtually residually finite rationally solvable (for instance whenever the mapping torus is hyperbolic), then $G$ is not coherent. This applies, in particular, when the image is a purely pseudo--Anosov free subgroups of the mapping class group.