Group cocycles and the ring of affiliated operators. (English) Zbl 1227.22003
The authors study cocycles of discrete countable groups with values in \(\ell^2G\) and the ring of affiliated operators \(\mathcal{U}G\). They clarify properties of the first cohomology of a group \(G\) with coefficients in \(\ell^2 G\) and answer several questions from Y. de Cornulier, R. Tessera and A. Valette [Transform. Groups 13, No. 1, 125–147 (2008; Zbl 1149.22006)]. Moreover, they obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first \(\ell^2\)-Betti number. They give numerous applications and examples of groups which satisfy their assumptions.
Reviewer: Francisco Pérez Acosta (La Laguna)
MSC:
| 22-XX | Topological groups, Lie groups |
| 20-XX | Group theory and generalizations |
| 43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
Keywords:
cocycles; discrete countable groups; ring; first cohomology; free subgroups; first \(\ell^2\)-Betti numberCitations:
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