The applications of Bieri-Neumann-Strebel invariant on K\"ahler groups

TL;DR

This paper explores the use of the Bieri-Neumann-Strebel invariant in the context of K"ahler groups, offering new proofs and insights into their algebraic properties.

Contribution

It provides a simpler proof of a known theorem and characterizes amenable K"ahler groups using the BNS invariant.

Findings

Simplified proof of Napier-Ramachandran theorem

Amenable K"ahler groups have empty BNS complement

Enhanced understanding of K"ahler group structures

Abstract

We give several applications of the Bieri-Neumann-Strebel invariant on K\"ahler groups. Specifically, we provide simpler proof of the Napier-Ramachandran theorem on the HNN extension about K\"ahler groups and show that amenable K\"ahler groups have an empty complement of the BNS invariant.