BNS Invariants and Algebraic Fibrations of Group Extensions

TL;DR

This paper investigates the algebraic fibrations of group extensions using BNS invariants, showing conditions under which such groups fiber and applying results to surface bundles to address a question of Hillman.

Contribution

It establishes new criteria for algebraic fibrations of group extensions based on BNS invariants and applies these to surface bundle groups to determine noncoherence.

Findings

Groups with $b_1(G) > b_1( ext{base}) > 0$ algebraically fiber.

Surface bundle groups with certain Albanese dimensions are noncoherent.

Answers a question of Hillman regarding surface bundles.

Abstract

Let $G$ be a finitely generated group that can be written as an extension \[ 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \] where $K$ is a finitely generated group. By a study of the BNS invariants we prove that if $b_1(G) > b_1(\Gamma) > 0$, then $G$ algebraically fibers, i.e. admits an epimorphism to $\Bbb{Z}$ with finitely generated kernel. An interesting case of this occurrence is when $G$ is the fundamental group of a surface bundle over a surface $F \hookrightarrow X \rightarrow B$ with Albanese dimension $a(X) = 2$. As an application, we show that if $X$ has virtual Albanese dimension $va(X) = 2$ and base and fiber have genus greater that $1$, $G$ is noncoherent. This answers for a broad class of bundles a question of J. Hillman.