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 $\rho :F_2\to Out(K)$ contains an element $\phi \in Out(K)$ such that the mapping torus $K{⋊}_{\phi }\mathbb{Z}$ 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.

This paper has been withdrawn. The content of this paper is subsumed in successive joint work with Robert Kropholler and Genevieve Walsh, see arXiv:2005.01202 [math.GR]

References in corpus (2)

• Cubulating hyperbolic free-by-cyclic groups: the general case
• Some noncoherent, nonpositively curved Kähler groups