BNS Invariants and Algebraic Fibrations of Group Extensions

Abstract

Let $G$ be a finitely generated group that can be written as an extension \[ 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} Γ\longrightarrow 1 \] 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 $\mathbb{Z}$ with finitely generated kernel. An interesting case of this occurrence is when $G$ is the fundamental group of a surface bundle over a surface $F\hookrightarrow X\to B$ 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.

arXiv commentFinal version, with some typo corrected. To appear in J. Inst. Math. Jussieu