  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.
