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BNS INVARIANTS AND ALGEBRAIC FIBRATIONS OF GROUP EXTENSIONS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 22 / Issue 2 / March 2023
- Published online by Cambridge University Press:
- 21 September 2021, pp. 985-999
- Print publication:
- March 2023
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- Article
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Let G be a finitely generated group that can be written as an extension
where K is a finitely generated group. By a study of the Bieri–Neumann–Strebel (BNS) invariants we prove that if
$$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \end{align*} $$
$b_1(G)> b_1(\Gamma ) > 0$
, then G algebraically fibres; that is, 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 fibre have genus greater that
$1$
, G is noncoherent. This answers for a broad class of bundles a question of J. Hillman ([9, Question 11(4)]). Finally, we show that there exist surface bundles over a surface whose BNS invariants have a structure that differs from that of Kodaira fibrations, determined by T. Delzant.
