  Given a Kaehler group $G$ and a primitive class $\phi \in H^1(G;Z)$, we show
that the rank gradient of $(G;\phi)$ is zero if and only if Ker $\phi$ is
finitely generated. Using this approach, we give a quick proof of the fact
(originally due to Napier and Ramachandran) that Kaehler groups are not
properly ascending or descending HNN extensions. Further investigation of the
properties of Bieri-Neumann-Strebel invariants of Kaehler groups allows us to
show that a large class of groups of orientation-preserving PL homeomorphisms
of an interval, which generalize Thompson's group $F$, are not Kaehler.
