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Group cocycles and the ring of affiliated operators

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In this article we study cocycles of discrete countable groups with values in ℓ 2 G and the ring of affiliated operators \(\mathcal{U}G\). We clarify properties of the first cohomology of a group G with coefficients in ℓ 2 G and answer several questions from De Cornulier et al. (Transform. Groups 13(1):125–147, 2008). Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first ℓ 2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.

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Authors and Affiliations

  1. Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Jesse Peterson

  2. University of Leipzig, Johannisgasse 26, 04103, Leipzig, Germany

    Andreas Thom

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  1. Jesse Peterson
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  2. Andreas Thom
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Correspondence to Andreas Thom.

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Peterson, J., Thom, A. Group cocycles and the ring of affiliated operators. Invent. math. 185, 561–592 (2011). https://doi.org/10.1007/s00222-011-0310-2

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