Homological Sigma-invariants are defined for locally compact Hausdorff groups and shown to recover and extend the classical discrete-group invariants.
Computer-assisted methods in Sigma-theory
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We develop an algorithm for recognizing whether a character belongs to $\Sigma^m$. In order to apply it we just need to know that the ambient group is of type $\mathrm{FP}_m$ or of type $\mathrm{F}_2$ and that the word problem is solvable for this group. Then finite data is sufficient proof of membership in $\Sigma^m$, not just for the given character but also for a neighborhood of it.
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Geometric invariants of locally compact groups: the homological perspective
Homological Sigma-invariants are defined for locally compact Hausdorff groups and shown to recover and extend the classical discrete-group invariants.