Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yongqiang Liu, Yuan Liu

TL;DR

This work generalizes the tropical approach to BNSR invariants to integral coefficients, producing finite unions of rational cones on the character sphere via Trop$_{\mathbb{Z}}$-tropicalizations of jump ideals and extending the comparison with tropical varieties to $\mathbb{Z}$-coefficients. It proves a central inclusion linking $\Sigma^k(X; \mathbb{Z})$ to $\mathrm{S}(\mathrm{Trop}_{\mathbb{Z}}(\mathcal{J}^{\le k}(X; \mathbb{Z})))^c$, with a second inclusion using field coefficients, and shows dense rational points in these cones. The paper then applies these results to Kähler groups, characterizing when weighted right-angled Artin groups can be Kähler and establishing a metabelian reduction for $\Sigma^1$ in this class. It also develops the consequences for Dwyer–Fried sets and presents concrete examples illustrating strict versus equality cases, highlighting the interplay between integral tropical geometry and classical group invariants. Overall, the work deepens the bridge between tropical geometry and topological finiteness properties, yielding both conceptual and computable tools for studying BNSR invariants in integral settings and for Kähler-group classifications.

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 Kähler groups. Specifically, we classify the Kähler 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.

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

TL;DR

This work generalizes the tropical approach to BNSR invariants to integral coefficients, producing finite unions of rational cones on the character sphere via Trop-tropicalizations of jump ideals and extending the comparison with tropical varieties to -coefficients. It proves a central inclusion linking to , with a second inclusion using field coefficients, and shows dense rational points in these cones. The paper then applies these results to Kähler groups, characterizing when weighted right-angled Artin groups can be Kähler and establishing a metabelian reduction for in this class. It also develops the consequences for Dwyer–Fried sets and presents concrete examples illustrating strict versus equality cases, highlighting the interplay between integral tropical geometry and classical group invariants. Overall, the work deepens the bridge between tropical geometry and topological finiteness properties, yielding both conceptual and computable tools for studying BNSR invariants in integral settings and for Kähler-group classifications.

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 Kähler groups. Specifically, we classify the Kähler 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.

Paper Structure

This paper contains 20 sections, 26 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Applications
  5. Organization
  6. Bieri, Groves and Strebel's results
  7. Connections with tropical geometry
  8. Tropical variety over a valued field
  9. Tropical variety over rings
  10. Proof of Theorem \ref{['thm:compare_two_bnsr']}
  11. The BNSR invariants revisited
  12. Jump loci ideal and Alexander ideal
  13. Proof of Theorem \ref{['thm:compare_two_bnsr']}
  14. Comparing with Bieri, Groves and Strebel's results
  15. Examples and application to Dwyer-Fried sets
  16. ...and 5 more sections

Key Result

Theorem 1.1

With the above notations and assumptions, we have Moreover, $\mathrm{S}({\mathrm{Trop}}_{\mathbb{Z}}(\mathcal{J}^{\leqslant k}(X; \mathbb{Z})))$ and $\mathrm{S}({\mathrm{Trop}}_{\mathbbm{k}}(\mathcal{J}^{\leqslant k}(X; \mathbbm{k})))$ are both finite unions of rationally defined convex cones over polyhedrons on the sphere $\mathrm{S}(G)$. In parti

Figures (2)

  • Figure 1: Comparison of several tropicalizations.
  • Figure 2: \ref{['example 2']}.

Theorems & Definitions (69)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3: Weighted right-angled Artin groups
  • Remark 1.4
  • Theorem 1.5: DPS, Corollary 11.14
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.9
  • Theorem 2.1: BS80, Proposition 2.1
  • Remark 2.2
  • ...and 59 more