Sigma-invariants and tropical varieties

Abstract

The Bieri-Neumann-Strebel-Renz invariants Σ q (X, Z) ⊂ H 1 (X, R) of a connected, finitetype CW-complex X are the vanishing loci for Novikov-Sikorav homology in degrees up to q, while the characteristic varieties V q (X) ⊂ H 1 (X, C × ) are the nonvanishing loci for homology with coefficients in rank 1 local systems in degree q. We show that each BNSR invariant Σ q (X, Z) is contained in the complement of the tropical variety associated to the algebraic variety V ≤q (X), and provide applications to several classes of groups and spaces.