BNS Invariants and Algebraic Fibrations of Group Extensions

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.