Group cocycles and the ring of affiliated operators
By J. Peterson, Andreas Thom. 2007
143 citations recorded by the source corpus.
In this article we study cocycles of discrete countable groups with values in ℓ2G and the ring of affiliated operators $\mathcal{U}G$. We clarify properties of the first cohomology of a group G with coefficients in ℓ2G 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 ℓ2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/eec0bab2b3f4316eaec5cd548db82b154f73de87
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Bass-Serre rigidity results in von Neumann algebras (2008, background): “are infinite. Note that i and �j have a vanishing first L 2 -Betti number (see [5, 28 ]).”
- A random matrix approach to the Peterson-Thom conjecture (2020, background): “In [57], Peterson-Thom proved various indecomposability and malnormality results for groups with positive first L(2)-Betti number.”
- ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, background): “The phenomenon that the usage of algebras of unbounded operators affiliated with the group von Neumann algebra simplifies algebraic matters in the theory of L(2)-invariants has already been used successfully in [62].”
- On cocycle superrigidity for Gaussian actions (2009, background): “Since a non-amenable group has vanishing first `2-Betti number if and only if it has vanishing first cohomology into its left regular representation [3, 31], we derive the following corollary.”