Sigma-invariants and tropical varieties

The Bieri-Neumann-Strebel-Renz invariants $\Sigma^q(X,$\mathbb{Z}$)\subset H^1(X,$\mathbb{R}$)$ofaconnected,finite-typeCW-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 ${\mathrm{\Sigma }}^q(X,\mathbb{Z})$ 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.