  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.
