  We describe some of the connections between the Bieri-Neumann-Strebel-Renz
invariants, the Dwyer-Fried invariants, and the cohomology support loci of a
space X. Under suitable hypotheses, the geometric and homological finiteness
properties of regular, free abelian covers of X can be expressed in terms of
the resonance varieties, extracted from the cohomology ring of X. In general,
though, translated components in the characteristic varieties affect the
answer. We illustrate this theory in the setting of toric complexes, as well as
smooth, complex projective and quasi-projective varieties, with special
emphasis on configuration spaces of Riemann surfaces and complements of
hyperplane arrangements.
