Mathematics – Group Theory
Scientific paper
2008-12-14
Proceedings of the London Mathematical Society 100 (2010), no. 3, 795-834
Mathematics
Group Theory
40 pages; accepted for publication in the Proceedings of the London Mathematical Society
Scientific paper
10.1112/plms/pdp045
We investigate the relationship between the geometric Bieri-Neumann-Strebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as K\"ahler and quasi-K\"ahler manifolds.
Papadima Stefan
Suciu Alexander I.
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