Incoherence and fibering of many free-by-free groups

TL;DR

This paper introduces a homological criterion called excessive homology to identify incoherence in many free-by-free groups, including hyperbolic and non-hyperbolic cases, and explores algebraic fibering properties.

Contribution

It establishes a new homological criterion for incoherence and applies it to broad classes of free-by-free groups and related groups, advancing understanding of their algebraic structure.

Findings

Free-by-free groups with excessive homology are incoherent.

Finite index subgroups of $F_2\rtimes F_n$ are incoherent.

Groups with excessive homology algebraically fiber.

Abstract

We show that free-by-free groups satisfying a homological criterion, which we call excessive homology, are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply this criterion to finite index subgroups of $F_2\rtimes F_n$ to show incoherence of all such groups, and to other similar classes of groups. Furthermore, we show that a large class of groups, including free-by-free, surface-by-surface, and finitely generated-by-RAAG, algebraically fiber if they have excessive homology.