Sigma-invariants and tropical varieties

TL;DR

This paper explores the relationship between Bieri-Neumann-Strebel-Renz invariants and tropical varieties, revealing their containment relations and applications to various groups and spaces.

Contribution

It establishes a connection between BNSR invariants and tropical varieties, providing new insights into their geometric and algebraic properties.

Findings

BNSR invariants are contained in the complement of associated tropical varieties.

The results apply to multiple classes of groups and topological spaces.

Provides a new geometric perspective on algebraic invariants.

Abstract

The Bieri-Neumann-Strebel-Renz invariants $\Sigma^q(X,\mathbb{Z})\subset H^1(X,\mathbb{R})$ of a connected, finite-type CW-complex $X$ are the vanishing loci for Novikov-Sikorav homology in degrees up to $q$, while the characteristic varieties $\mathcal{V}^q(X) \subset H^1(X,\mathbb{C}^{\times})$ are the nonvanishing loci for homology with coefficients in rank 1 local systems in degree $q$. We show that each BNSR invariant $\Sigma^q(X,\mathbb{Z})$ is contained in the complement of the tropical variety associated to the algebraic variety $\mathcal{V}^{\le q}(X)$, and provide applications to several classes of groups and spaces.