A Spectral Sequence to Compute L-betti Numbers of Groups and Groupoids Roman Sauer and Andreas Thom
نویسنده
چکیده
We construct a spectral sequence for L-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations.
منابع مشابه
A Hochschild-serre Spectral Sequence for Extensions of Discrete Measured Groupoids Roman Sauer and Andreas Thom
We construct a Hochschild-Serre spectral sequence for L-type cohomology groups of discrete measured groupoids using a blend of tools from homological algebra and ergodic theory. Applications include a theorem about existence of normal amenable subrelations in measured equivalence relations, results about possible stabilizers of measure preserving group actions, and vanishing results or explicit...
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There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II 1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück’s dimension theory, Gaboriau’s definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition...
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