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

Abstract

Let $G$ be the semidirect product $\Gamma ⋊F_2$ where $\Gamma$ is either the free group $F_n$, $n>1$ or the fundamental group $S_g$ of a closed surface of genus $g>1$. 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. Although many groups have been shown to be incoherent using virtual algebraic fibering, we also show that not every free-by-free group virtually algebraically fibers.

There is an error in the proofs of Theorem 3.1 and Theorem 3.5 which we cannot immediately fix. We thank Hongbin Sun for pointing this error out. We are merging the results on groups which do not virtually algebraically fiber (which are unaffected by this error) with our paper arXiv:2103.06930

References in corpus (5)

• Cubulating hyperbolic free-by-cyclic groups: the irreducible case
• Subgroup classification in Out(F_n)
• Some noncoherent, nonpositively curved Kähler groups
• Cubulating Surface-by-free Groups
• Free by cyclic groups are large