Rank gradients of infinite cyclic covers of Kaehler manifolds

TL;DR

This paper investigates the properties of Kaehler groups, showing that their rank gradients vanish under certain conditions and proving they are not certain types of HNN extensions, with implications for groups related to Thompson's group F.

Contribution

It provides a new criterion linking rank gradients to the finite generation of kernels in Kaehler groups and applies this to exclude certain groups from being Kaehler.

Findings

Rank gradient of (G;φ) is zero iff Ker φ is finitely generated.

Kaehler groups are not properly ascending or descending HNN extensions.

Certain groups related to PL homeomorphisms of an interval are not Kaehler.

Abstract

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.