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

Rank gradients of infinite cyclic covers of Kaehler manifolds