Virtually RFRS Mapping Tori and Coherence

Abstract

Let G be a finitely presented group that can be written as an extensionwhere 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 ρ : F 2 → Out(K) contains an element ϕ ∈ Out(K) such that the mapping torus K ⋊ ϕ 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.