Incoherence of free-by-free and surface-by-free groups

Abstract

Let G be the semidirect product Γ ¸F2 where Γ is either the free group Fn, n ą 1 or the fundamental group Sg of a closed surface of genus g ą 1. We prove that G is incoherent, solving two problems posed by D. Wise. This implies an affirmative answer to a question of J. Hillman on the fundamental group of a surface bundle over a surface.