Bieri-Neumann-Strebel-Renz invariants and tropical varieties of integral homology jump loci

TL;DR

This paper extends the connection between Bieri-Neumann-Strebel-Renz invariants and tropical varieties of homology jump loci from complex to integral coefficients, providing new bounds and applications to Kähler groups.

Contribution

It generalizes Suciu's results to integral coefficients and offers improved bounds for BNSR invariants, with applications to classifying certain Kähler groups.

Findings

Translated positive-dimensional components are detectable via tropical varieties.

Provides a better upper bound for BNSR invariants.

Classifies Kähler groups within weighted right-angled Artin groups.

Abstract

Papadima and Suciu studied the relationship between the Bieri-Neumann-Strebel-Renz (short as BNSR) invariants of spaces and the homology jump loci of rank one local systems. Recently, Suciu improved these results using the tropical variety associated to the homology jump loci of complex rank one local systems. In particular, the translated positive-dimensional component of homology jump loci can be detected by its tropical variety. In this paper, we generalize Suciu's results to integral coefficients and give a better upper bound for the BNSR invariants. Then we provide applications mainly to K\"ahler groups. Specifically, we classify the K\"ahler group contained in a large class of groups, which we call the weighted right-angled Artin groups. This class of groups comes from the edge-weighted finite simple graphs and is a natural generalization of the right-angled Artin groups.