TITLE: Finitely generated subgroups with infinite cyclic quotient
QUESTION [12 upvotes]: Suppose that $G$ is a finitely presented group and $H$ is a finitely generated normal subgroup such that $G/H$ is infinite cyclic. Is it true that $H$ is finitely presented?

REPLY [5 votes]: It worth noticing that the Sigma-invariants (also known as BNS- or BNSR-invariants) provide a strategy to answer such a question. 
Given a group $G$ of type $F_n$ and an integer $k \leq n$, the $k$-th Sigma-invariant $\Sigma^k(G)$ is a (complicated) subset of the character sphere
$$S(G):= \left( \mathrm{Hom}(G,\mathbb{R}) \backslash \{ 0 \} \right) / \text{positive scaling}$$
of $G$. Notice that the class of every non trivial morphism $G \to \mathbb{Z}$ is an element of $S(G)$.
Theorem: The kernel of $\chi : G \twoheadrightarrow \mathbb{Z}$ is of type $F_k$ if and only if $[\chi], [-\chi] \in \Sigma^k(G)$.
So the case $k=2$ corresponds to $\mathrm{ker}(\chi)$ finitely presented. Unfortunately, these invariants are usually quite difficult to compute. Nevertheless, there exists a useful method, based on Bestvina and Brady's Morse theory, extending their work on right-angled Artin groups (mentioned in Henry's answer). 
For instance, all the Sigma-invariants are completely known for right-angled Artin groups and some Thompson-like groups. An application I really like is:
Theorem: Any finitely presented normal subgroup of Thompson's group $F$ is of type $F_{\infty}$.
A few references on the subjet:

Strebel, Notes on the Sigma-invariants.
Bux & Gonzales, The Bestvina-Brady construction revisited - Geometric computation of $\Sigma$-invariants for right-angled Artin groups.
Witzel & Zaremsky, The $\Sigma$-invariants of Thompson's group $F$ via Morse theory.
Zaremsky, On the $\Sigma$-invariants of generalized Thompson groups and Houghton groups.
Zaremsky, Symmetric automorphisms of free groups, BNSR-invariants, and finiteness properties.
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